What is a set?
Leonard Blackburn
(I was sometimes given a bit more detail, but this was the crux of the
definition). I later read similar definitions by famous mathematicians
that refined this definition, giving what we generally call the
axiom schema of full comprehension, and I learned that these definitions
were invalidated by the discovery of Russell's Paradox (and other
paradoxes). Then I learned about axiomatic set theory. First, I was
excited because I thought: "Finally, I will get an airtight definition
of the word _set_." But after reading a little book on set theory by
Charles Pinter, the first chapter of _Set Theory_ by Thomas Jech, and
most recently, the first two chapters of _Set Theory_ by Frank Drake
(as well as a paper on the Continuum Hypothesis by Carol Karp and a
couple of other papers), it slowly dawned on me that I still don't know
what a set is! The axioms of ZFC do not provide me with a definition,
they merely attempt to say true things about some objects called _sets_
which are left undefined. Drake says:
"We cannot expect complete agreement on the question: what
are sets? But we can pick out certain things about which
there is agreement that these _are_ sets (whatever else may
be) and study these." (page 1 in text mentioned above)
He then continues to describe informally "the cumulative type structure
of sets" and he gives informal arguments that each of the axioms of ZFC
are true about this structure. His Platonism is vivid and exciting to me.
However, the picture of what a set is remains incomplete in my mind.
Does anyone know where I can read more about exactly what sets are? Have
there been attempts to refine the description of the cumulative type
structure of sets since Drake's book.
This reminds me of recursive functions. Before the numerous equivalent
definitions of recursive functions were given around the '30s, mathematicians
conceived of "functions computable by an algorthm." Before there was
a formal definition agreed upon, people could point to certain functions
and say that they were algorithmic, and there would be agreement. Then
eventually, a class of functions (the partial recursive functions) was
born and people agreed that this was a good definition of an algorithmic
function, and, indeed, much evidence was gathered in support of this
definition.
Could something like this ever happen with sets? Is it possible that
one day we might agree on a definition of set? Or is this a bad analogy,
perhaps because of the foundational nature of set theory?
Also, every professor I've ever had in college and now in graduate school
has used the word "set" frequently in his or her lectures. Does this mean
that they are all Platonists? In fact, I have never met a student or
professor of mathematics who doesn't talk about sets. Were they all Platonists?
Or is talk about sets sometimes just abbreviated talk about well-formed
formulas of set theory?
Furthermore, in every set theory book I have looked at (the ones
listed above as well as a book by Kunen, and a couple of others), there
is always some talk about models of ZFC, where by model one means
a structure that has a set as an underlying universe and a binary relation
on that set that interprets the membership relation symbol. This bothers
me for two reasons. (1) We don't know if ZFC has _any_ models at all.
It might be an inconsistent theory. So, why make statements about models
if it is possible that our whole conversation is empty? (2) I am
uncomfortable that the notion of model relies on the notion of set in the
first place. It seems illogical to interpret the theorems of set theory
(ZFC) in a set (equipped with a binary relation, which is itself a set)
itself. Maybe not illogical, but certainly strange.
Am I mistaken about something, or perhaps underinformed?
One more thing. In all of the set theory books mentioned above, the proofs
are always informal in the following sense. No deductive calculus is ever used
to establish that there is a formal deduction (as in Enderton's book,
_A Mathematical Introduction to Logic_) of the theorems at hand. Instead,
it is imagined that there is a model of ZFC, and an ordinary mathematical
proof is given with that model in mind. Does this mean that all of these
set theorists are Platonists?
Thanks in advance for any helpful comments.
Leonard Blackburn
Student, University of Minnesota
Paul Holbach
> news:<aa503d8.02092...@posting.google.com>...
> However, the picture of what a set is remains incomplete in my mind.
Everybody knows the famous definition by Cantor:
A set is a collection of things
(called the members or elements) ,the collec-
tion being regarded as a single object."Any
collection into a whole M of definite and sep-
arate objects m of our intuition or our thought."
(by G.Cantor in Contributions to the Founding
of the Theory of Transfinite Numbers
New York:Dover Publications,inc.)
But there also is an alternative definition by Cantor which is much
less known:
In the Gesammelte Abhandlungen (Collected Papers), p. 204, Cantor
writes the following in note 1:
"Unter einer 'Mannigfaltigkeit' oder 'Menge' verstehe ich nämlich
allgemein jedes Viele, welches sich als Eines denken lässt, d.h. jeden
Inbegriff bestimmter Elemente, welcher durch ein Gesetz zu einem
Ganzen verbunden werden kann; und ich glaube, hiermit etwas zu
definieren, was verwandt ist mit dem platonischen 'eidos' oder 'idea',
wie auch mit dem, was Platon in seinem Dialoge 'Philebos oder das
höchste Gut' 'mikton' nennt."
"By 'manifold' or 'set' I generally understand any many which may be
conceived of as one, ie any aggregate* of determinate elements which
can be united into a whole by some law; and I believe that, herewith,
I define something which is related to the Platonic 'eidos' or 'idea'
and also to what is called 'mikton' ('mixture',
'mingling'/'mingledness') by Plato in his dialogue 'Philebos and the
highest good'."
(translated by myself)
(*) "Aggregate" here seems the most adequate translation of
"Inbegriff".
This quotation shows clearly that Cantor is a steadfast Platonist!
But that doen´t mean at all that set theory must necessarily be
interpreted Platonistically!
If one simply regards sets as extensions of logical predicates - as I
do -, then one can well do without Platonism. Those specific
extensions are all the things sharing some property, ie there are two
sides of the very same coin:
1) The intensional side of a set: S = {x|Fx}
eg: S = {x|(0<x<10)&x in N} = {x|(natural number larger than 0 but
smaller than 10)x}
2) The extensional side of a set: S = {x1,x2,x3, ...}
eg (see 1): S = {1,2,3,4,5,6,7,8,9}
Correspondingly, extensionality is a property of concepts indicating
the number of objects that possess the set-defining property in
question. For example, the concept "planet in our solar system" is
realized nine times, ie there are nine objects that instantiate the
property of being a planet in our solar system, ie there are nine
objects to which the predicate "planet in our solar system" correctly
applies. Correspondingly, the set of all planets in our solar system
has the cardinality 9. 'Nine-numberedness' is the property of the
concept "planet in our solar system".
There are many entities which share some properties and there are many
linguistic predicates which can be correctly ascribed to those
entities by a community of language-users. There´s no need for
Platonic 'ideas' which are nothing but figments of our imagination!
By the way, you might find the following text interesting:
http://home.uchicago.edu/~wwtx/cantor.pdf
regards
PH
Mike Oliver
> Then I learned about axiomatic set theory. First, I was
> excited because I thought: "Finally, I will get an airtight definition
> of the word _set_."
The hope of making things "airtight" is a character trait that
tends to lead people to interest in the foundations of mathematics.
That was the way I felt too, long ago. I've come to the conclusion
that it's hopeless, and really not so desirable anyway. It's
turtles all the way down -- and that's OK.
To get a mental concept of what a set is, I think the best thing
is to start small; let's talk about what a set of natural numbers
is. We'll just slide right on past the question of what a
natural number itself is -- that's a lower turtle than the one
we're looking at. The von Neumann coding of natural numbers
as sets is just that, a coding; it's not going to tell you anything
about what the naturals are that you didn't already know.
So is it clear to you what a set of naturals is, without
trying to generalize to what a set is, period? I mean,
you understand that we're talking about completely *arbitrary*
ways of gathering naturals together, that there does
not need to be any rule saying *how* they're gathered,
and that what we're abstracting out is just *which* natural
numbers they contain, not *why* they contain them (that is,
they're extensional rather than intensional collections).
If that's all clear, then you need to visualize *all* such
sets being gathered into a completed whole, the powerset
of the naturals. Then we can start moving up the ladder
in a way I'm sure you're familiar with.
There is one more piece: You have to make friends with
the notion of wellordering as an *informal* concept, and
understand informally why any two wellorderings are of
comparable length. Then you abstract those lengths to
get an informal notion of "ordinal". This is what keeps
the picture from being circular -- the stages of the construction
are the ordinals, the real, informal ordinals, not just
the ones you find in the object being constructed itself.
Now, back to the turtles: None of this needs to strike
you a priori as being 100% convincing. That's not its
job. It just needs to be clear enough to let you suspend
disbelief and work with it. There's always time to
come back and reexamine pieces, after you know more about
how the objects behave.
Mitchell Smith
Leonard Blackburn wrote:
> Does this mean that all of these
> set theorists are Platonists?
>
The distinction between Platonists and anything else is murky since formalists and
intuitionists simply bury similar assumptions in different places. The problem is
that they invent metalanguages and other contexts which obsfucate solutions.
What follows is the first exposition of my personal research in class theory. It is
excerpted from a personal correspondence, so there may be some odd statements
interspersed within it.
In a classical sense, a set is a sort of class. However, my own questions have to
do with faulty assumptions in metamathematics. I do not operate under the paradigm
of conventional model theory, although my sentences are formulated in first-order
logic.
You may assume that I am talking about sets--I am certainly talking about the
interpretation of Venn diagrams, which is more than you get in a formal set theory
course (or in Jech).
Do not expect this to be easy, although the mathematics is trivial.
You will have to do the Venn diagrams yourself. They do not seem to post well.
But, I am new to newsgroup participation.
I will try again right here:
---------------------------------------
| |
| ----------------- |
| | | |
| | x * | |
| | | |
| ----------------- |
| X |
----------------------------------------
x in X
---------------------------------------
| |
| ----------------- |
| | | |
| | a * | |
| | | |
| ----------------- |
| b * |
----------------------------------------
not (a = b) [meaning a and b are topologically separated]
I begin with circularly referenced definitions to obtain predicates with
properties. I do not begin with objects. The Venn diagram components suffice for
interpretation. In order to understand why, you will have to read the philosophy of
Immanuel Kant with regard to mathematical propositions arising from a priori
synthetic cognitions.
The universal quantifier is used in anticipation of an intensional interpretation
through the definition for a maximal class. The initial issue is to establish an
identity predicate based on topological separation rather than the axiom of
extensionality.
Normally, the universal quantifier cannot have a reference in the language because
the universal quantifier is treated as a parameter instantiated with a class
interpreted with respect to the membership relation. My satisfaction strategy uses
a lax order relation which you may think of as set inclusion, although its actual
implementaion is the mereology-theoretic "part" relation.
There is an explicit assumption of a property called almost universality which
provides for interpreting the maximal class as a collection (with respect to
membership)
If you are willing to explore, I will explain as much as I can and direct you to
references.
No one understands what I am doing here. Let me begin by pointing out that I never
considered Russell's Paradox to be a state of reality. In my eyes, it is nothing
more than an example of illiteracy.
Had they focused on defining their terms instead of inventing formal systems....
You may find a link for mereotopology at
http://www.acsu.buffalo.edu/~ryankohl/1996Smith.pdf
where you can get intuition concerning how the "proper_part" predicate can have an
interpretation whch is not set-theoretic.
Look carefully at axiom AIP5 and theorem TIP1. Just remember that the author is
also using a metalinguistic identity predicate in his definition of the universe.
But, you should compare this theorem to the Generic Model Theorem and Cohen's
forcing language.
In fact, since you have Jech, look at the section on Boolean-valued models and
compare it to my definitions. You will find that it is membership and order which
are substantive--not identity.
Good luck. Feel free to contact me. I can practically assure you that this will
satisfy your intutions.
:-)
mitch smith
---------------------------------------------------------
I will try to provide you with the appropriate rigor here.
Strictly interpreted, the natural language term "language primitive" only
has an intensional meaning in conventional first-order model theory when its
referent is among the undefined symbols which comprise the language of the
model. Defined symbols do not fall within the scope of that reference.
However, consideration must be given to the fact that the language
generated here is being formulated using circular definitions. Were it not
for the fact that my paradigm is attempting to reasonably introduce a
reference for the class universe in the formal language, my sentences could
easily conform to interpretation with respect to a conventional model. In
that case, the language primitives would be "proper_part" and "in."
The definition for "proper_part" is given by:
for all a for all b( a proper_part b iff (
forall c( b proper_part c implies a proper_part c )
and
exists c( a proper_part c and not (b proper_part c) ) ) )
The definition for "in" is given by:
for all a for all b( a in b iff (
for all c( b proper_part c implies a in c )
and
exists c( a in c and not (b proper_part c) ) ) )
In order to understand the parallelism of these definitions, it it
necessary to consider the naive use of Venn diagrams with respect to a
representation of "x in X,"
--------------------------------------------------------
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|
|
|
|
------------------ |
| |
| |
| |
| |
| | x *
| |
| |
| |
|
------------------ |
|
X |
|
|
|
|
|
|
|
|
--------------------------------------------------------
The Venn diagram has two types of representations which shall be referred to
as "dot" and "circle" representations. The referent of x is a "dot"
representation and the referent of X is a "circle" representation.
With regard to object reference in an arbitrary first-order model, the
constituents of the model domain are represented using "dot"
representations. The only admissible "circle" representation is the one
representing the model domain, or universe of discourse. Moreover, since
this usage reflects parameterization of the universal quantifier, the
"circle" representation manifests itself in the metalanguage only. If one
wishes to interpret properties differentiating objects as inducing
separations of the model domain, one is implicitly invoking a topological
extension to the model and, thereby, is no longer working in a first-order
context.
Considering these observations, we infer that "dot" representations reflect
object semantics while "circle" representations reflect collection
semantics. Thus, with respect to the syntactic form of "x in X" and its
naive representation using Venn diagrams, we conclude that the "in" predicate
relates objects to collections.
The parallelism in the definitions above reflect an extremely careful
implementation of the relationship between "dot" representations and "circle"
representations. This differentiation is important because hereditarily
defined classes are the unique object type which present both object
semantics and collection semantics.
The predicate symbols of a conventional first-order model relate references
of similar object type to one another. Given that the "proper_part"
predicate is defined with the expectation of relating collections to
collections, its introduction is consistent with this standard practice.
In constrast, the "in" predicate is normally part of the metalanguage where
its intensional meaning conveys the well-established naive usage discussed
above. When it appears in the formal language of a conventional first-order
model, the object type of the model is necessarily that of hereditarily
defined classes. However, it cannot enforce that interpretation in isolation
because the object type of its left argument is ambiguous. That is why its
formal description requires a previously defined language element whose
arguments are consistent and unambiguous.
While the model constituents of Zermelo-Fraenkel set theory may be
hereditarily defined classes, the same cannot be said for arbitrary
first-order models. Consequently, the conventional model-theoretic approach
interprets the "in" predicate with respect to different contexts. To my
knowledge, no one has ever attributed Russell's paradox to poor context
management. In my humble opinion, the analysis presented here is far more
suggestive than the naive conclusion that the universal class is "too big."
In any case, the two definitions above exhibit parallel syntactic
structures so that a clear distinction between object semantics and
collection semantics is expressed through the syntactic form of expressions
used to formulate the theory.
You are already familiar with the "equiv" predicate:
for all a for all b( a equiv b iff (
for all c ( a proper_part c iff b proper_part c )
and
for all c ( c proper_part a iff c proper_part b )
and
for all c ( a in c iff b in c )
and
for all c ( c in a iff c in b ) )
This is not a language primitive because it is defined in terms of previously
introduced symbols. Nevertheless, you may wish to compare it to the use of
the '=' as a logical symbol of the language in conventional first-order model
theory.
Your first inclination will be to claim that this is a language primitive
just like the one found there. After all, my definition is formulated to
correspond with the axioms of identity.
Well conventional model theory generally holds that '=' is a logical symbol
of the language. To say that is to assert that it has a fixed interpretation
in all models with respect to the *given* model domain. Thus, it is not a
primitive symbol subject to non-standard interpretations. That is,
conventional language primitives are supposed to be indeterminate. In
contrast, interpretation for '=' is undetermined only to the extent that the
universal quantifier has not been instantiated.
I am more than willing to entertain the idea that '=' is a language
primitive provided that we have an expectation of some terminology that
differentiates between symbol-types according to the special rules governing
their usage. Realistically, however, all I require is the understanding that
usage assumptions arising from familiarity with conventional model theory may
not be justified.
In terms of the paradigm I am trying to establish, this is just another
definition. However, because of its syntactic structure and its relationship
to the symbols introduced using circular reference, the equivalence relation
it defines may be designated as the characteristic equivalence relation for
the language.
The reason this definition corresponds with the usual formulation of the
axioms of identity is because a language needs an equivalence relation which
governs the intensional interpretation of terms from the language having no
variables to their referents. When Quine wrote "Set Theory and Its Logic,"
it was apparent that definitions such as this were ambiguous:
"The sense of 'x=y' given by the plan of definition
illustrated in (i) and (ii) may or may not really
be identity;"
But, this deficiency could be easily corrected because the universal
quantifier need only be instantiated over a domain comprised of the
equivalence classes of 'x=y.' Yet, once implemented, this adjustment leaves
the referent of a constant term with a multiplicity different from one.
In my humble opinion, that does not conform with common intuitions
concering the precise language usage of mathematicians. Well-defined
constant bindings should resolve to individuals, not collections.
Whether done explicitly through an assignment or implicitly through a
metalinguistic identity predicate, the extensional interpretation of '=' as a
logical symbol in conventional model theory is being done with respect to the
diagonal of a specific model domain. That is not the case with this
equivalence relation.
The equivalence relation defined above suffices for the purpose of
introducing terms into the language. That is its only function. It is not a
language primitive because it is derived from previously introduced language
symbols.
I can now turn to what you really wanted to see. The specific axioms and
theorems of my construction.
The three axioms are given by:
1. Axiom of Conjoinment
for all a for all b(
for all c( c proper_part a iff c proper_part b )
implies
for all c( a in c iff b in c ) )
2. Axiom of Constituency
for all a for all b(
for all c( c in a implies c in b )
implies
for all c( b proper_part c implies a proper_part c ) )
3. Axiom of Power
for all a exists b for all c(
c in b iff c proper_part a )
Through these axioms and the theorems deduced below the sequence of
implications,
for all c( a proper_part c iff b proper_part c )
implies
\forall c( c in a iff c in b )
implies
\forall c( c proper_part a iff c proper_part b )
implies
\forall c( a in c iff b in c )
implies
\forall c( a proper_part c iff b proper_part c )
will be demonstrated thereby establishing
a equiv b iff for all c ( a proper_part c iff b proper_part c )
a equiv b iff for all c ( c proper_part a iff c proper_part b )
a equiv b iff for all c ( a in c iff b in c )
a equiv b iff for all c ( c in a iff c in b )
as theorems.
Observe that the third implication from the sequence is directly obtained
from the first axiom. This is reasonable given that we are working without a
model specification. A violation of the identity predicate shall correspond
to an inconsistency in the axiomatization.
Now for the theorems.
Assertion:
for all x for all y(
x proper_part y
implies
for all z( z in x implies z in y))
Proof:
Let x and y be arbitrary such that x proper_part y. Moreover, let z be
arbitrary such that z in x. This establishes the premise.
From
for all a for all b( a in b iff (
for all c( b proper_part c implies a in c )
and
exists c( a in c and not (b proper_part c) ) ) )
it follows that for every p, x proper_part p implies z in p as z in x was
given in the premise.
Thus z in y since x proper_part y.
This completes the proof.
Assertion:
for all x for all y(
for all z( y proper_part z implies x proper_part z)
implies
for all z( z in x implies z in y))
Proof:
Consider how the order relation was introduced into the language,
for all a for all b( a proper_part b iff (
for all c( b proper_part c implies a proper_part c )
and
exists c( a proper_part c and not (b proper_part c) ) ) )
The negation of an atomic formula asserting this relation generates three
mutually exclusive cases. Namely,
for all c( b proper_part c implies a proper_part c )
and
for all c( a proper_part c implies b proper_part c )
exists c( b proper_part c and not (a proper_part c) )
and
for all c( a proper_part c implies b proper_part c )
exists c( b proper_part c and not (a proper_part c) )
and
exists c( a proper_part c and not (b proper_part c) )
Note that the last two cases are of the form (A and B) and (A and not B).
This is important because the next step of this proof will be directed toward
eliminating any possibility that the first case might hold under the
conditions of the premise. The remainder of the proof will then capitalize
on this simple relationship between the remaining two cases.
To begin, let x and y be arbitrary. In addition, assume the contrapositive
assertion of the theorem statement. That is, suppose
exists z( z in x and not (z in y) )
holds. This establishes the conditions of the premise.
Now, the contrapositive assertion of the previous theorem gives
for all x for all y(
exists z( z in x and not (z in y) )
implies
not (x proper_part y) )
So, not (x proper_part y) holds as this follows from the premise.
Now, the formula,
for all k( y proper_part k implies x proper_part k )
and
for all k( x proper_part k implies y proper_part k )
is an equivalent alphabetic variant for the first case of not (x proper_part
y)
as discussed above.
It is not possible for this formula to hold simultaneously with those of the
premise. Assumption of this statement will lead to the incorrect conclusion
that every element of x is an element of y.
To see this, let z in x be as in
exists z( z in x and not (z in y) )
From the definition
for all a for all b( a in b iff (
for all c( b proper_part c implies a in c )
and
exists c( a in c and not (b proper_part c) ) ) )
it follows that for every p, x proper_part p implies z in p.
Now the same relationship must be shown to hold for z and y.
Let q be such that y proper_part q. From
for all k( y proper_part k implies x proper_part k )
and
for all k( x proper_part k implies y proper_part k )
it follows that x proper_part q. Consequently, z in q by virtue of the
first theorem.
Thus, for every q, y proper_part q implies z in q.
Next,
for all a for all b( a in b iff (
for all c( b proper_part c implies a in c )
and
exists c( a in c and not (b proper_part c) ) ) )
also implies that there is some p such that z in p and not x proper_part p.
As before, it is necessary to show a corresponding relationship for z and y.
Using
for all k( y proper_part k implies x proper_part k )
and
for all k( x proper_part k implies y proper_part k )
it is clear that not (y proper_part p) is implied by not (x proper_part p).
Consequently, for any p that satisfies z in p and not (x proper_part p) it
must also be true that z in p and not (y proper_part p).
Thus, using an equivalent alphabetic variant and combining with the previous
result,
for all q( y proper_part q implies z in q )
and
exists q( z in q and not (y proper_part q) ) ) )
is satisfied and z in y must be consequent to z in x.
But, z was chosen to satisfy
exists z( z in x and not (z in y) )
whence it cannot be true that z in y.
Consequently, the case of not (a proper_part b) expressed by,
for all c( b proper_part c implies a proper_part c )
wedge
for all c( a proper_part c implies b proper_part c )
must lead to contradiction if it is applied to the situation in the premise.
This completes the first step of the proof.
As previously observed, the remaining cases for not (a proper_part b) have
the
form (A and B) and (A and not B). The subformula which holds in
both cases is given by
exists c( b proper_part c and not (a proper_part c) )
In the present circumstance,
exists z( y proper_part z and not (x proper_part z) )
is the equivalent alphabetic variant, and,
not (x proper_part y)
implies
exists z( y proper_part z and not (x proper_part z) )
must hold.
Thus,
for all x for all y (
exists z( z in x and not (z in y) )
implies
exists z( y proper_part z and not (x proper_part z) ) )
is proved.
The statement of the theorem follows by contraposition. This completes the
proof.
Assertion:
for all x for all y( x proper_part y iff (
for all z( z in x implies z in y )
and
exists z( z in y and not (z in x) ) )
Proof:
Recalling the second axiom given above,
for all a for all b(
for all c( c in a implies c in b )
implies
for all c( b proper_part c implies a proper_part c ) )
we see that it is the converse of
for all a for all b(
forall c( b proper_part c implies a proper_part c )
implies
for all c( c in a implies c in b ) )
But the equivalent alphabetic variant,
for all x for all y(
for all z( y proper_part z implies x proper_part z )
implies
for all z( z in x implies z in y ) )
asserts the previous theorem.
Consequently, the logical equivalences
for all x for all y(
for all z( z in x implies z in y )
iff
for all z( y proper_part z implies x proper_part z ) )
for all x for all y(
exists z( z in y and not (z in x) )
iff
exists z( x proper_part z and not (y proper_part z) ) )
hold.
Recalling the definition for the order relation,
for all a for all b( a proper_part b iff (
for all c( b proper_part c implies a proper_part c )
and
exists c( a proper_part c and not (b proper_part c) ) ) )
it is obvious how the assertion of the theorem may be obtained by appropriate
substitutions with respect to the subformulas of its definiendum using the
equivalences just realized.
These substitutions yield
for all x for all y(
x proper_part y iff (
for all z( z in x implies z in y )
and
exists z( z in y and not (z in x) ) ) )
This completes the proof.
Assertion:
for all x for all y(
for all z( z in x implies z in y )
implies
for all z( z proper_part x implies z proper_part y ) )
Proof:
Let x and y be arbitrary. Suppose further that h in x implies h in y
for every class h. This establishes the premise.
Now let k be such that k proper_part x By the previous theorem,
for all a for all b (
a proper_part b implies (
for all c ( c in a implies c in b)
and
exists c( c in b and not (c in a) ) )
So, q in k implies q in x for every class q. It follows that q in y
from the implication stated in the premise.
That is, for every q, q in k implies q in y.
Also by the previous theorem, k proper_part x implies that there is some q
for which q in x and not (q in k). This implies q in y as in the
previous step.
So, q in y and not (q in k) holds for some q.
Thus,
for all k for all y (
for all q( q in k implies q in y )
and
exists q( q in y and not (q in k) ) )
whence k proper_part y.
Since k was arbitrarily chosen to satisfy k proper_part x, the statement,
for all x for all y (
for all z( z in x implies z in y )
implies
for all z( z proper_part x implies z proper_part y ) )
must hold.
This completes the proof.
Assertion:
for all x for all y (
for all z( x in z implies y in z)
implies
for all z( x proper_part z implies y proper_part z ) )
Proof:
Let x and y be arbitrary. Assume the premise for the contrapositive
statement
of the theorem. That is, suppose there is some h such that x proper_part h
and not (y proper_part h). This establishes the premise for the proof.
From the third axiom,
for all a exists b for all c (
c in b
iff
c proper_part a )
it follows that there must exist some k such that for all q, q proper_part h
if and only if q in k.
Then x in k since x proper_part h from the premise. Similarly, not (y in k)
since not (y proper_part h) from the premise.
Thus, using equivalent alphabetic variants, the assertion
for all x for all y (
exists z( x proper_part z and not (y proper_part z) )
implies
exists z( x in z and not (y in z) ) )
must hold.
The assertion of the theorem is obtained by formulating the contrapositive
statement.
This completes the proof.
The sentences,
for all a for all b (
a equiv b
iff
for all c ( a proper_part c iff b proper_part c ) )
for all a for all b (
a equiv b
iff
for all c ( c proper_part a iff c proper_part b ) )
for all a for all b (
a equiv b
iff
for all c ( a in c iff b in c ) )
for all a for all b (
a equiv b
iff
for all c ( c in a iff c in b ) )
now follow as theorems.
At this point, the identity predicate may be defined in terms of that
characterization which supports an interpretation of distinctness relative to
the topological separation of points. The formal definition is given by
for all a for all b (
a = b
iff
for all c ( a in c iff b in c )
Since the usual notion of extensionality is available as a theorem, Venn
diagrams may now be naively interpreted. It may, perhaps, be constructive to
now consider (a \= b)
--------------------------------------------------------
|
|
|
|
|
------------------ |
| |
| |
| |
| |
| | a *
| |
| |
| |
|
------------------ |
|
c |
|
|
|
|
| b
* |
|
|
--------------------------------------------------------
so that the formulation of this predicate is clear. Equivalence is
established with respect to definable properties while distinctness is
established with respect to topological separation relative to interpretation
of the universe of discourse as a discrete topology.
Nevertheless, there is a problem with the construction presented thus far.
Namely, a referent has not yet been assigned to the universal quantifier.
With the predicates now available, however, it is a simple matter to
introduce a greatest class and a least class.
References to the greatest class and the least class are introduced into
the language with the definitions,
for all a ( a = V() iff (
for all b( b proper_part a xor a = b ) ) )
for all a ( a = 0() iff (
for all b( a proper_part b xor b = a ) ) )
The reader should be able to verify that the descriptions in the definienda
are well defined. Note, however, that a definition must be substantiated
through a proof that some object satisfies the definiendum or by virtue of an
axiom asserting existence:
exists a( for all b( b proper_part a xor a = b ) )
exists a( for all b( a proper_part b xor b = a ) )
It is interesting to observe that these language terms have been introduced
using the mereology-theoretic "proper_part" relation. As noted above, it is
the "in" relation which governs model-theoretic object semantics. This
problem is rectified through the axiom of almost universality,
for all a(
exists b( a proper_part b )
implies
exists b( a \in b ) )
This axiom provides the precise intuition for characterizing the maximal
class as a universe of discourse,
Assertion:
exists a( for all b( b in a xor a = b) )
Proof:
Indeed, let a=V. Then, if not (b = V), b proper_part V
by the assumption of V as a maximal class. The axiom of almost
universality implies the existence of a class, c, such that
b in c. Now, either c = V or c proper_part V. In the former
case, b in V is immediate. If the latter case holds, b in V
follows from the theorem,
for all x for all y (
x proper_part y
iff (
for all z( z in x implies z in y )
and
exists z( z in y and not (z in x) ) )
Now, suppose that not (b in V). From the definition of the
"in" predicate, either there exists c such that V proper_part c
and not (b in c) or for every c it follows that not (b in c) or
V proper_part c. The first case is simply not possible since it
contradicts the definition of V as a maximal class. Similarly,
not (b in c) for every c must be concluded from the second case.
But, if not (b in c) for every c, then the contrapositive of the
axiom of almost universality implies not (b proper_part c) for every c. It
follows that b = V.
The theorem is complete.
It should be clear from this exposition that a description of a universal
class requires the use of a partial order relation. Antisymmety must
generate an equivalence relation whose associated equivalence classes have a
multiplicity of one. With regard to this constraint, the mereology-theoretic
"proper_part" relation may be thought of as a metalinguistic relation with
respect to the class-theoretic "in" relation.
Conversely, the universe of discourse instantiating the universal
quantifier of a first-order language is naively understood as a collection.
With regard to this constraint, the class-theoretic "in" relation may be
thought of as a metalinguistic relation with respect to the
mereology-theoretic "proper_part" relation.
In the end, connectivity associated with quantification is the
implementation issue rather than a specific platonistic entity. It manifests
itself with dual interpretations--namely, an existential quantifier and a
universal quantifier. Any formulation that successfully describes the class
universe must both reflect this duality and be responsible for it.
John
I asked a noted set theorist:
> If you have a moment: Are ZF and NBG *about* classes and
> membership, or are they uninterpreted formalisms? (This
> question has been bothering me a lot.)
His answer:
> I'm afraid I can't save you from being bothered by this one! There's
> no right answer. If you catch someone using ZF and NBG, you can
> ask them in particular whether they are talking about classes and
> membership, or just using an uninterpreted formalism. You might
> sometimes get a coherent answer. But not from von Neumann, who
> (if I read him right) took the view (a) that set theory is
> about the universe of sets and (b) that there is no universe
> of sets.
For a study in evasion, by someone not so high in the Q-Moolah-
Tive Hierarchy, see Torkel Franzen's contributions to the
thread, "Is ZFC axiomatizable in FOPC", in fa.analytic-philosophy.
--John
Aatu Koskensilta
Leonard Blackburn wrote:
> When I was younger, I was told that a set was a "collection of objects"
> (I was sometimes given a bit more detail, but this was the crux of the
> definition). I later read similar definitions by famous mathematicians
> that refined this definition, giving what we generally call the
> axiom schema of full comprehension, and I learned that these definitions
> were invalidated by the discovery of d othRussell's Paradox (aner
> paradoxes). Then I learned about axiomatic set theory. First, I was
> excited because I thought: "Finally, I will get an airtight definition
> of the word _set_." But after reading a little book on set theory by
> Charles Pinter, the first chapter of _Set Theory_ by Thoma Jesch, and
> most recently, the first two chapters of _Set Theory_ by Frank Drake
> (as well as a paper on the Continuum Hypothesis by Carol Karp and a
> couple of other papers), it slowly dawned on me that I still don't know
> what a set is! The axioms of ZFC do not provide me with a definition,
> they merely attempt to say true things about some objects called _sets_
> which are left undefined.
This is the case with all mathematical objects. What is a function? What
is a natural number?
> Drake says:
>
> "We cannot expect complete agreement on the question: what
> are sets? But we can pick out certain things about which
> there is agreement that these _are_ sets (whatever else may
> be) and study these." (page 1 in text mentioned above)
>
> He then continues to describe informally "the cumulative type structure
> of sets" and he gives informal arguments that each of the axioms of ZFC
> are true about this structure. His Platonism is vivid and exciting to me.
> However, the picture of what a set is remains incomplete in my mind.
>
> Does anyone know where I can read more about exactly what sets are? Have
> there been attempts to refine the description of the cumulative type
> structure of sets since Drake's book.
There have been some attempts to derive the axioms of ZFC from certain
natural axioms pertaining the cumulative hierarchy. However, I don't
think you can ever have a "definition of set" in the sense you seem to
use. Sets are quite like functions. A function is an object for which
the application relation is defined and is "functional", nothing else
can be said about them. Similarly sets are objects for which the
membership relation is somehow given.
Even natural numbers have this property. What is a natural number? I
don't know of any good answer. However, we know what sorts of relations
natural numbers have with each other (and other objects).
> This reminds me of recursive functions. Before the numerous equivalent
> definitions of recursive functions were given around the '30s, mathematicians
> conceived of "functions computable by an algorthm." Before there was
> a formal definition agreed upon, people could point to certain functions
> and say that they were algorithmic, and there would be agreement. Then
> eventually, a class of functions (the partial recursive functions) was
> born and people agreed that this was a good definition of an algorithmic
> function, and, indeed, much evidence was gathered in support of this
> definition.
> Could something like this ever happen with sets? Is it possible that
> one day we might agree on a definition of set? Or is this a bad
> analogy,
> perhaps because of the foundational nature of set theory?
But what is a function? It's merely something that can be applied to
other things. The class of recursive functions is a subclass of the
class of all functions, and it's precisely what makes them different
from other functions (belonging to the closure of some function
formation operations, or being computable by Turing machine or being
lambda definable, &c) that you consider important. But this is merely
defining a *subclass* of the itself "undefined" (in the sense I infer
you use) class of functions.
>
> Also, every professor I've ever had in college and now in graduate school
> has used the word "set" frequently in his or her lectures. Does this mean
> that they are all Platonists? In fact, I have never met a student or
> professor of mathematics who doesn't talk about sets. Were they all Platonists?
> Or is talk about sets sometimes just abbreviated talk about well-formed
> formulas of set theory?
Talk about sets does not commit one to platonism (and the alternative is
not extreme silly formalism).
> Furthermore, in every set theory book I have looked at (the ones
> listed above as well as a book by Kunen, and a couple of others), there
> is always some talk about models of ZFC, where by model one means
> a structure that has a set as an underlying universe and a binary relation
> on that set that interprets the membership relation symbol. This bothers
> me for two reasons. (1) We don't know if ZFC has _any_ models at all.
> It might be an inconsistent theory.
Well, in simlar vain it is possible that PA is inconsistent.
We have it in a sense backwards these days. Arguable for natural
numbers, and to a much lesser extent for sets, the original intention of
the axiomatisation was to characterise precisely (up to isomorphism)
what the structure we study is and what properties it haves by virtue of
being just that structure. Thus the structure comes first, and its
axiomatic description comes later. I find it rather odd that people go
about insisting that there is some real possibility that ZFC or PA is
inconsistent. They're not, they have models, namely natural numbers and
the cumulative hierarchy. (Of course, we could all be mistaken about
everything, but what is that supposed to mean in the first place?).
> So, why make statements about models
> if it is possible that our whole conversation is empty? (2) I am
> uncomfortable that the notion of model relies on the notion of set in the
> first place. It seems illogical to interpret the theorems of set theory
> (ZFC) in a set (equipped with a binary relation, which is itself a set)
> itself. Maybe not illogical, but certainly strange.
> Am I mistaken about something, or perhaps underinformed?
Your conception is certainly shared by most people who have superficial
and sometimes even non-superficial familarity with the incompleteness
results of logic. I personally favour the model theoretic view of world;
logic is the study of structures and associated languages, answering
such question as whether the language is rich enough to describe the
structure, what sort of properties leave a certain fragment, &c.
I find the view that logic is somehow the "foundation" of mathematics
rather odd; one needs quite an elaborate mathematical machinery to even
get to the interesting results of mathematical logic. Yet this view is
what underlies for example your conception of the role of logic. It is
perfectly reasonable from a normal mathematical point of view to use set
theory (as an informal theory of the cumulative hierarchy) to study the
metatheory of axiomatic set theory. From what other perspective should
we approach the study?
(As a side note, forcing and many other important results in axiomatic
set theory can be rendered into a form that perhaps pleases you more; it
is possible to view these as proof transformations. It is in general
possible, though not sensible, to interprete model theoretic result as
results about classes of languages and theories without reference to
models).
> One more thing. In all of the set theory books mentioned above, the proofs
> are always informal in the following sense. No deductive calculus is ever used
> to establish that there is a formal deduction (as in Enderton's book,
> _A Mathematical Introduction to Logic_) of the theorems at hand. Instead,
> it is imagined that there is a model of ZFC, and an ordinary mathematical
> proof is given with that model in mind. Does this mean that all of these
> set theorists are Platonists?
No, it means that they, as do virtually all mathematicians, realise that
there is no realistic hope to ever produce a mathematical work of
substance rendered in formal first order logic.
> Thanks in advance for any helpful comments.
>
> Leonard Blackburn
> Student, University of Minnesota
--
Aatu Koskensilta (aatu.kos...@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
Mitchell Smith
John wrote:
> > But not from von Neumann, who
> > (if I read him right) took the view (a) that set theory is
> > about the universe of sets and (b) that there is no universe
> > of sets.
>
> For a study in evasion, by someone not so high in the Q-Moolah-
> Tive Hierarchy, see Torkel Franzen's contributions to the
> thread, "Is ZFC axiomatizable in FOPC", in fa.analytic-philosophy.
>
> --John
Set theory is about the universe of sets. Contrary to the apparent problem posed by Russell's paradox, it is possible
to formulate a meaningful intensional interpretation of the universal quantifier. Von Neumann made his statement with
respect to a context of formal systems whose metamathematical assumptions are in error.
For the last two weeks I have been trying to engage discussion of this matter. You may find my original attempt in
sci.math.research under "What distinguishes identity among equivalence relations?"
Because I am new to newsgroups and because I have been hesitant about undue claims, my original post does nothing more
than present a list of sentences in TeX syntax. If, however, you examine the responses to my post carefully, you will
note that few of the authors even bothered to demonstrate any curiosity.
In contrast, my replies attempted to return the discussion back on point by discussing the failure of conventional
model theory to provide a model which could be interpreted as the class universe. There were no attempts to
understand the issues I was attempting to convey. I only succeeded in voluntary committment to refrain from posts to
sci.math.research because my posts were too long.
I have made numerous long posts on sci.math and sci.logic, hoping, to find individuals whose curiosity would motivate
their participation. Instead, I find that people simply avoid complexity they cannot understand immediately.
Moreover, they have otherwise convinced themselves that the early researchers in foundational mathematics could not
have been in error.
The only person who has maintained a sustained interest in my formal sentences was an active newsgroup participant to
whom I will give credit if ever he so desires. However, he failed to understand that my formal sentences were
designed to attack faulty metamathematical assumptions. Consequently, he verified the logic of my proofs in order to
conclude that my system introduced no "new" mathematics.
Of course, it should not. The initial assumptions of any prudent person (for a discussion of what constitutes a
prudent person in my estimation, look under sci.logic, "Re:Re:What is a crackpot") should take Zermelo-Fraenkel set
theory at face value and focus on the issues which prevent description of a model for the class universe--that is,
metamathematical assumptions.
If you have any interest in any of this, I will gladly explain. But, the paradigm is based on circular reference,
making it quite difficult when one has not been exposed to it before. Moreover, it is predicate-based rather than
object-based. The "objects" of the system arise from Venn diagrams and their existence is justified in terms of the
Kantian paradigm of a priori synthetic cognitions.
In any case, the entry point from the paradigm of formal systems is Von Neumann's observation and the understanding
that the identity predicate must be understood in terms of topological separation of points rather than
extensionality.
:-)
mitch smith
Mitchell Smith
Aatu Koskensilta wrote:
> No, it means that they, as do virtually all mathematicians, realise that
> there is no realistic hope to ever produce a mathematical work of
> substance rendered in formal first order logic.
Twenty years ago, I asked myself how the formal description--that is, a formal
definition--of the identity predicate would be implemented. Fifteen years ago, I
began the thought experiments which produced such a definition using circular
reference. I interpret this usage as being equivalent to natural language usage since
my predicates have properties regardless of extensional interpretation.
I have spent the last twelve years trying to understand how my sentences should be
interpreted. The trivial statements are easily accommodated by conventional model
theory. The problem was that I saw a different way to formulate a satisfaction
strategy in terms of order relations. By so doing, I could introduce a reference for
a maximal class into the language.
Consideration of this problem, however, is extremely deep. Ultimately, it brought me
to the conclusion that the fundamental objects of mathematics are topologies rather
than collections. If you compare the group theory axioms with the set theory axioms,
it is clear that all groups satisfy the group axioms, but, the axioms of
Zermelo-Fraenkel set theory primarily describe closure properties on the class
universe. So, what object-type is the class universe, and how are we to understand
that object-type with regard to inheritance by its subclasses.
Collection semantics arises from the fact that the axiom schema of separation
precludes language users from proving that the subclasses of the class universe are
not totally disconnected. On the other hand, the paradox associated with the
comprehension schema prevents separations of the class universe. So, as a topology,
the class universe is connected. Collection semantics for the class universe follow
from a property called almost universality.
This leaves a simple idea. The class universe is a connected topology for which every
proper subspace is totally disconnected. The next step is to restructure the axioms
to reflect this.
Look for my posts with formal elements. Then decide for yourself concerning whether
or not my hopes are realistic.
:-)
mitch smith
Mark
>When I was younger, I was told that a set was a "collection of
There's nothing more complicated to it than that.
Every finite collection of size 2 or more is definitely a set.
Sizes 0 and 1 have not historically been regarded as constituting
bona fide collections. But naive set theory makes the distinction
anyhow, between x and {x}; and allows for {}.
There is not universal disagreement on whether {0,1,2,...} is a
set. Cantor postulated this to be a set. An (increasilgly)
widely known alternative to Cantor's postulate has emerged and
forms the basis of Euclidean Set Theory, based on one of
Euclid's Common Notions (in Book I):
The whole is greater than its parts.
Such an axiom, formulated in the form:
Euclidean Axiom:
f: S -> S is one-to-one => f is onto
yields a counter-postulate to Cantor's Axiom of Infinity.
What's suprising is just how much you can STILL do within
that framework, and there is (in fact) a wide range of
activity centering on the Euclidean version of set theory
in recent times (e.g. attempts at developing a Euclidean
version of real analysis).
Historically, the notion of a collection regardable as
a single entity was denoted by the older meaning and
connotation of the word "number" (e.g., "a number of horses
were seen in the field" means "a collection of horses...").
The notion of an EXTENSION of a predicate was considered
to be something quite different. So, what the axiom of
comprehension defined
extension-of-p = { x: p(x) }
is really a quite different kind of animal than what
you call a "collection". This is more properly thought of
as a class, and a class was not generally considered as
being the same thing as a single unifiable collection.
The problem, of course, is in confusing these two concepts.
That leads straight to contradiction.
There's lots of other set theories. The ZFC & von Neumann
alternative jibes with the notion that a "class is a really-really-
big-set" and work on that angle. The NF version, in contrast,
has such an entity as a universal set. Its non-sets are those
classes whose defining conditions are too complex or which
violate well-formed typing conventions.
The axioms you typically see (those below) work just as well
in ZFC or Euclidean set theory. All of these properties are
well-motivated for finite collections:
Object = individual or set.
An individual is any object (other than {}) which has no members.
Pairs: {u,v} is a set for any objects u, v.
Empty set: {} is a set.
Restricted comprehension: { u in A: p(u) } is a set, if A is (with
p(x) usually placed under some restrictions).
Restricted comprehension doesn't hold in NF.
Replacement: { f(u): u in A } is a set if A is (with restrictions
usually placed on what functions f(x) are allowed).
The two that are a little harder to motivate for finite collections,
but which are still perfectly viable are:
Union: The union of a set of sets is a set.
Power set: P(A) = { x: x subset of A } is a set if A is.
The main issue comes when you get to Cantor's axiom of infinity.
And prior to Cantor, this would have been flat-out rejected (and
was still treated by his peers with skepticism even at the time).
It also creates weird synergistic effects when combined with the
power set axiom. In particular, you get well-known anomalies BOTH
when you assume the Axiom of Choice is true AND when you assume
it's false.
Leonard Blackburn
[snip]
> But there also is an alternative definition by Cantor which is much
> less known:
>
> In the Gesammelte Abhandlungen (Collected Papers), p. 204, Cantor
> writes the following in note 1:
> "Unter einer 'Mannigfaltigkeit' oder 'Menge' verstehe ich nämlich
> allgemein jedes Viele, welches sich als Eines denken lässt, d.h. jeden
> Inbegriff bestimmter Elemente, welcher durch ein Gesetz zu einem
> Ganzen verbunden werden kann; und ich glaube, hiermit etwas zu
> definieren, was verwandt ist mit dem platonischen 'eidos' oder 'idea',
> wie auch mit dem, was Platon in seinem Dialoge 'Philebos oder das
> höchste Gut' 'mikton' nennt."
>
> "By 'manifold' or 'set' I generally understand any many which may be
> conceived of as one, ie any aggregate* of determinate elements which
> can be united into a whole by some law; and I believe that, herewith,
> I define something which is related to the Platonic 'eidos' or 'idea'
> and also to what is called 'mikton' ('mixture',
> 'mingling'/'mingledness') by Plato in his dialogue 'Philebos and the
> highest good'."
> (translated by myself)
> (*) "Aggregate" here seems the most adequate translation of
> "Inbegriff".
Thank you for this definition.
>
> This quotation shows clearly that Cantor is a steadfast Platonist!
>
> But that doen´t mean at all that set theory must necessarily be
> interpreted Platonistically!
> If one simply regards sets as extensions of logical predicates - as I
> do -, then one can well do without Platonism.
But aren't there only countably many logical predicates because
there are only finitely many symbols in our language? Thus, this
would seem to imply that there are only countably many sets.
Wouldn't we like to believe that there are uncountably many sets?
(Maybe I am misunderstanding, but in any case I still don't see how
one can do without Platonism.)
> Those specific
> extensions are all the things sharing some property, ie there are two
> sides of the very same coin:
> 1) The intensional side of a set: S = {x|Fx}
> eg: S = {x|(0<x<10)&x in N} = {x|(natural number larger than 0 but
> smaller than 10)x}
> 2) The extensional side of a set: S = {x1,x2,x3, ...}
> eg (see 1): S = {1,2,3,4,5,6,7,8,9}
>
> Correspondingly, extensionality is a property of concepts indicating
> the number of objects that possess the set-defining property in
> question. For example, the concept "planet in our solar system" is
> realized nine times, ie there are nine objects that instantiate the
> property of being a planet in our solar system, ie there are nine
> objects to which the predicate "planet in our solar system" correctly
> applies. Correspondingly, the set of all planets in our solar system
> has the cardinality 9. 'Nine-numberedness' is the property of the
> concept "planet in our solar system".
> There are many entities which share some properties and there are many
> linguistic predicates which can be correctly ascribed to those
> entities by a community of language-users. There´s no need for
> Platonic 'ideas' which are nothing but figments of our imagination!
>
> By the way, you might find the following text interesting:
>
> http://home.uchicago.edu/~wwtx/cantor.pdf
Thanks for this reference. I will look at it.
>
> regards
> PH
Leonard Blackburn
> The hope of making things "airtight" is a character trait that
> tends to lead people to interest in the foundations of mathematics.
> That was the way I felt too, long ago. I've come to the conclusion
> that it's hopeless, and really not so desirable anyway. It's
> turtles all the way down -- and that's OK.
I too am beginning to see that it is hopeless, but I don't see why
it wouldn't be desirable.
>
> To get a mental concept of what a set is, I think the best thing
> is to start small; let's talk about what a set of natural numbers
> is. We'll just slide right on past the question of what a
> natural number itself is -- that's a lower turtle than the one
> we're looking at. The von Neumann coding of natural numbers
> as sets is just that, a coding; it's not going to tell you anything
> about what the naturals are that you didn't already know.
good point. Now we're being Platonic--assuming that naturals exist
independent of a formalization. Is that right?
>
> So is it clear to you what a set of naturals is, without
> trying to generalize to what a set is, period? I mean,
> you understand that we're talking about completely *arbitrary*
> ways of gathering naturals together, that there does
> not need to be any rule saying *how* they're gathered,
> and that what we're abstracting out is just *which* natural
> numbers they contain, not *why* they contain them (that is,
> they're extensional rather than intensional collections).
I think my intuitive understanding of a set of natural
numbers is not quite as good as my intuitive understanding of the set
of all natural numbers, which is not quite as good as my intuitive
understanding of the number 17.
>
> If that's all clear, then you need to visualize *all* such
> sets being gathered into a completed whole, the powerset
> of the naturals. Then we can start moving up the ladder
> in a way I'm sure you're familiar with.
Yes. But way up high on the ladder, the steps get hazy.
>
> There is one more piece: You have to make friends with
> the notion of wellordering as an *informal* concept, and
> understand informally why any two wellorderings are of
> comparable length. Then you abstract those lengths to
> get an informal notion of "ordinal". This is what keeps
> the picture from being circular -- the stages of the construction
> are the ordinals, the real, informal ordinals, not just
> the ones you find in the object being constructed itself.
Yes! This makes sense. I didn't think of the possibility
of "real" ordinals before. In order to understand the cumulative
hierarchy of sets I need to first understand what an ordinal is.
>
> Now, back to the turtles: None of this needs to strike
> you a priori as being 100% convincing. That's not its
> job. It just needs to be clear enough to let you suspend
> disbelief and work with it. There's always time to
> come back and reexamine pieces, after you know more about
> how the objects behave.
Thanks.
Leonard Blackburn
> The distinction between Platonists and anything else is murky since formalists and
> intuitionists simply bury similar assumptions in different places. The problem is
> that they invent metalanguages and other contexts which obsfucate solutions.
>
This is good to hear. I've been suspecting lately that there is very
little difference between a Platonist and a non-Platonist. We probably
all have almost isomorphic beliefs when it comes to basic concepts in
mathematics, we just use different language to talk about the same things.
> What follows is the first exposition of my personal research in class theory.
I look forward to reading it when I have more time.
kise...@mindspring.com
Start with the empty set 0. There is a sentences that defines it:
a) there exists an empty set -- Ex Vy not (y e x)
b) the empty set is unique -- if (Vz not (z e x) and Vz not (z e y)
then x=y
Then start creating more sets from this:
1 is the set {0}, 2 is the set {0,1}, 3 is the set {0,1,2}
w is the set of a natural numbers - dont remember how it is defined --
probably something like the following:
y is the predecessor of x if x={y}. w is the intersection of all sets
that satisfy the property that each member has a predecessor except
for one element - that being the empty set.
So continue along and you can defined rationals, irrationals,
cardinals, ordinals or whatever.
So that is what a set is!
Leonard Blackburn
> This is the case with all mathematical objects. What is a function? What
> is a natural number?
Good point. But I was hoping that functions, numbers, and other
objects
are just sets. If that's not an aesthetic viewpoint, then at least I
would still be able to do mathematics using just sets as the only
objects.
> There have been some attempts to derive the axioms of ZFC from certain
> natural axioms pertaining the cumulative hierarchy. However, I don't
> think you can ever have a "definition of set" in the sense you seem to
> use. Sets are quite like functions. A function is an object for which
> the application relation is defined and is "functional", nothing else
> can be said about them. Similarly sets are objects for which the
> membership relation is somehow given.
>
> Even natural numbers have this property. What is a natural number? I
> don't know of any good answer. However, we know what sorts of relations
> natural numbers have with each other (and other objects).
>
> But what is a function? It's merely something that can be applied to
> other things. The class of recursive functions is a subclass of the
> class of all functions, and it's precisely what makes them different
> from other functions (belonging to the closure of some function
> formation operations, or being computable by Turing machine or being
> lambda definable, &c) that you consider important. But this is merely
> defining a *subclass* of the itself "undefined" (in the sense I infer
> you use) class of functions.
Ok, I see what you mean. While we have a definition that identifies
the recursive functions within the class of all functions, we don't
have a definition of the class of all functions. So, my analogy was
poor. In order to get a definition of a set somehow analogous to the
definition of a recursive function, I would need a larger given
context
to work in.
> > Or is talk about sets sometimes just abbreviated talk about well-formed
> > formulas of set theory?
>
> Talk about sets does not commit one to platonism (and the alternative is
> not extreme silly formalism).
Why doesn't talk about sets commit one to platonism? Can you
elaborate.
Are sets not mathematical objects? Are not mathematical objects
platonic
objects? I was thinking that when one utters a sentence about sets,
that
she is either making a statement about platonic objects or her
statement
is a translation of a well-formed formula of set theory. And I would
think
that the desire to translate formal set-theory sentences into talk
about objects indicates a platonic tendency.
> > me for two reasons. (1) We don't know if ZFC has _any_ models at all.
> > It might be an inconsistent theory.
>
> Well, in simlar vain it is possible that PA is inconsistent.
Yes, of course.
>
> We have it in a sense backwards these days. Arguable for natural
> numbers, and to a much lesser extent for sets, the original intention of
> the axiomatisation was to characterise precisely (up to isomorphism)
> what the structure we study is and what properties it haves by virtue of
> being just that structure.
But to think about the structure first is to be a platonist, isn't it?
> Thus the structure comes first, and its
> axiomatic description comes later. I find it rather odd that people go
> about insisting that there is some real possibility that ZFC or PA is
> inconsistent. They're not, they have models, namely natural numbers and
> the cumulative hierarchy.
But when I said that ZFC might be inconsistent I meant to make a very
precise and true statement. Namely, using the rules of a sound and
complete first-order predicate calculus (like the one I learned in
Enderton's _A Mathematical Introduction to Logic_) it may be possible
to derive from the axioms of ZFC a sentence and its negation. Are
you saying this isn't possible? There is no proof within ZFC that
ZFC is consistent, and the validity of any proof that ZFC is
consistent in a larger theory would depend on the consistency of that
larger theory.
So, I don't know what you mean by saying that ZFC and PA have models.
I grant you that it is hard to argue that our platonic structure of
the
natural numbers, about which we have much intuition, is not a model of
PA.
But I think our picture of the cumulative hierarchy of sets is much
more hazy. At least it is for me. I haven't seen a clear, complete,
and unambiguous definition of this hierarchy, even just in plain
English.
Also, doesn't the belief that ZFC and PA are consistent rely, then,
on the belief that these platonic structures exist and serve as
models?
And then, doesn't this mean that every mathematician is either a
platonist
or an extreme silly formalist who doesn't believe there is any meaning
behind mathematics? (Maybe I don't comprehend the definition of
'platonist'.)
> (Of course, we could all be mistaken about
> everything, but what is that supposed to mean in the first place?).
> Your conception is certainly shared by most people who have superficial
> and sometimes even non-superficial familarity with the incompleteness
> results of logic. I personally favour the model theoretic view of world;
> logic is the study of structures and associated languages, answering
> such question as whether the language is rich enough to describe the
> structure, what sort of properties leave a certain fragment, &c.
Does that make you a platonist? I want to be clear on the definition.
>
> I find the view that logic is somehow the "foundation" of mathematics
> rather odd; one needs quite an elaborate mathematical machinery to even
> get to the interesting results of mathematical logic. Yet this view is
> what underlies for example your conception of the role of logic. It is
> perfectly reasonable from a normal mathematical point of view to use set
> theory (as an informal theory of the cumulative hierarchy) to study the
> metatheory of axiomatic set theory. From what other perspective should
> we approach the study?
I don't know. But I feel the need that another perspective be
invented.
>
> (As a side note, forcing and many other important results in axiomatic
> set theory can be rendered into a form that perhaps pleases you more; it
> is possible to view these as proof transformations. It is in general
> possible, though not sensible, to interprete model theoretic result as
> results about classes of languages and theories without reference to
> models).
>
> > it is imagined that there is a model of ZFC, and an ordinary mathematical
> > proof is given with that model in mind. Does this mean that all of these
> > set theorists are Platonists?
>
> No, it means that they, as do virtually all mathematicians, realise that
> there is no realistic hope to ever produce a mathematical work of
> substance rendered in formal first order logic.
I guess I must be confused about the definition of 'platonism.' Do
some
of these mathematicians believe that there is hope to produce
substantial
mathematics within some other formal framework? If not, doesn't that
make them Platonists?
I really am starting to believe that all mathematicians are platonists
(and I don't intend that word to have a negative connotation). But
of course, this belief could be wiped out, if my understanding of the
word changes.
Thanks for all your articulate comments,
Leonard Blackburn
> blac...@math.umn.edu (Leonard Blackburn) writes:
> >When I was younger, I was told that a set was a "collection of
> >objects".
>
> There's nothing more complicated to it than that.
There must be! Otherwise, the collection of all sets that
do not contain themselves would be a set.
>
> Every finite collection of size 2 or more is definitely a set.
> Sizes 0 and 1 have not historically been regarded as constituting
> bona fide collections. But naive set theory makes the distinction
> anyhow, between x and {x}; and allows for {}.
>
> There is not universal disagreement on whether {0,1,2,...} is a
> set. Cantor postulated this to be a set. An (increasilgly)
> widely known alternative to Cantor's postulate has emerged and
> forms the basis of Euclidean Set Theory, based on one of
> Euclid's Common Notions (in Book I):
>
> The whole is greater than its parts.
>
> Such an axiom, formulated in the form:
>
> Euclidean Axiom:
> f: S -> S is one-to-one => f is onto
>
> yields a counter-postulate to Cantor's Axiom of Infinity.
If we accepted this counter-postulate, does that mean
that {0,1,2,...} cannot be a set because of the function
f(n)=n+1? Or does it mean that maybe the set exists but
not the function?
>
> What's suprising is just how much you can STILL do within
> that framework, and there is (in fact) a wide range of
> activity centering on the Euclidean version of set theory
> in recent times (e.g. attempts at developing a Euclidean
> version of real analysis).
This sounds very interesting.
>
> Historically, the notion of a collection regardable as
> a single entity was denoted by the older meaning and
> connotation of the word "number" (e.g., "a number of horses
> were seen in the field" means "a collection of horses...").
>
> The notion of an EXTENSION of a predicate was considered
> to be something quite different. So, what the axiom of
> comprehension defined
>
> extension-of-p = { x: p(x) }
>
> is really a quite different kind of animal than what
> you call a "collection". This is more properly thought of
> as a class, and a class was not generally considered as
> being the same thing as a single unifiable collection.
I see.
To what anomalies are you referring? I wasn't aware there existed
any anomalies, even when you assume all of ZFC.
Thanks for the comments,
John
> Leonard Blackburn wrote:
> > When I was younger, I was told that a set was a "collection of objects"
> > (I was sometimes given a bit more detail, but this was the crux of the
> > definition). I later read similar definitions by famous mathematicians
> > that refined this definition, giving what we generally call the
> > axiom schema of full comprehension, and I learned that these definitions
> > were invalidated by the discovery of d othRussell's Paradox (aner
> > paradoxes). Then I learned about axiomatic set theory. First, I was
> > excited because I thought: "Finally, I will get an airtight definition
> > of the word _set_." But after reading a little book on set theory by
> > Charles Pinter, the first chapter of _Set Theory_ by Thoma Jesch, and
> > most recently, the first two chapters of _Set Theory_ by Frank Drake
> > (as well as a paper on the Continuum Hypothesis by Carol Karp and a
> > couple of other papers), it slowly dawned on me that I still don't know
> > what a set is! The axioms of ZFC do not provide me with a definition,
> > they merely attempt to say true things about some objects called _sets_
> > which are left undefined.
>
> This is the case with all mathematical objects. What is a function?
>
> > Drake says:
> >
> > "We cannot expect complete agreement on the question: what
> > are sets? But we can pick out certain things about which
> > there is agreement that these _are_ sets (whatever else may
> > be) and study these." (page 1 in text mentioned above)
> >
> > He then continues to describe informally "the cumulative type structure
> > of sets" and he gives informal arguments that each of the axioms of ZFC
> > are true about this structure. His Platonism is vivid and exciting to me.
> > However, the picture of what a set is remains incomplete in my mind.
> >
> > Does anyone know where I can read more about exactly what sets are? Have
> > there been attempts to refine the description of the cumulative type
> > structure of sets since Drake's book.
>
> There have been some attempts to derive the axioms of ZFC from certain
> natural axioms pertaining the cumulative hierarchy. However, I don't
> think you can ever have a "definition of set" in the sense you seem to
> use. Sets are quite like functions. A function is an object for which
> the application relation is defined and is "functional", nothing else
> can be said about them. Similarly sets are objects for which the
> membership relation is somehow given.
What do you mean, SOMEHOW GIVEN? From the axioms governing "in", don't
the properties of this relations follow? And if so, when axioms
conflict--as (C3,C4) conflict with N4--isn't it so that at least one
of these is mistaken?
C3 EyAx[x in y <-> Et(x in t) & P(x)] (with y not free in P(x))
C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> x = y}]
N4 AxAy[Az(z in x <-> z in y) -> x = y]
If set theorists aren't talking about sets, haven't a whole lot of
us been taken?
>
> > Furthermore, in every set theory book I have looked at (the ones
> > listed above as well as a book by Kunen, and a couple of others), there
> > is always some talk about models of ZFC, where by model one means
> > a structure that has a set as an underlying universe and a binary relation
> > on that set that interprets the membership relation symbol. This bothers
> > me for two reasons. (1) We don't know if ZFC has _any_ models at all.
> > It might be an inconsistent theory.
>
> Well, in simlar vain it is possible that PA is inconsistent.
>
> We have it in a sense backwards these days. Arguable for natural
> numbers, and to a much lesser extent for sets, the original intention of
> the axiomatisation was to characterise precisely (up to isomorphism)
> what the structure we study is and what properties it haves by virtue of
> being just that structure. Thus the structure comes first, and its
> axiomatic description comes later.
If the structure comes first--and that's all that matters--what could
warrant (N1-N9) over (C1-C8), given that every model of one is also
a model of the other?
C1 AxAyAz[xIy -> (z in x <-> z in y)] LL1
C2 AxAyAz[Az(z in x <-> z in y) <-> (Az(x in z <-> y in z))] LL2
C3 EyAx[x in y <-> Et(x in t) & P(x)] (with y not free in P(x))
Classification
C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> xIy}]
Extensionality
C5 AaAbAy(Ax[x in y <-> (xIa v xIb)] -> yIy) Axiom of Pairs
C6 AaAy(Ax[x in y -> x in a] -> (aIa -> yIy)) Axiom of Subsets
C7 AaAy[Ax(x in y <-> Ew(w in a & x in w)) -> (aIa -> yIy)] Axiom of Unions
C8 Aa(Ay[Ax(x in y <-> Aw(w in x -> w in a))) -> (aIa -> yIy]) Axiom of Power
Sets
N1 AxAyAz[x=y -> (x in z <-> y in z)] LL 1
N2 AxAyAz[x=y -> (z in x <-> z in y)] LL 2
N3 EyAx[x in y <-> (Et(x in t) & P(x))] (with y not free in (x))
Classification
N4 AxAy[Az(z in x <-> z in y) -> x = y] Extensionality
N5 AaAb[set a & set b -> Ey(set y & Ax(x in y <-> x=a v x=b))] Axiom of Pairs
N6 Aa[set a -> Ay(Ax(x in y -> x in a) -> set y)] Axiom of Subsets
N7 Aa[set a -> Ey(set y & Ax(x in y <-> Ew(w in a & x in w)))]
Axiom of Unions
N8 Aa[set a -> Ey(set y & Ax(x in y <-> Aw(w in x -> w in a)))]
Axiom of Power Sets
N9 Ex(set x & Ay~(y in x)) Empty Set Axiom
> I find it rather odd that people go
> about insisting that there is some real possibility that ZFC or PA is
> inconsistent. They're not, they have models, namely natural numbers and
> the cumulative hierarchy. (Of course, we could all be mistaken about
> everything, but what is that supposed to mean in the first place?).
By the same token, haven't the experts perhaps been mistaken about the
reflexivity of identity?
--John
>
> >
Barb Knox
blac...@math.umn.edu (Leonard Blackburn) wrote:
> Aatu Koskensilta <aatu.kos...@xortec.fi> wrote in message
[snip]
> > Talk about sets does not commit one to platonism (and the alternative is
> > not extreme silly formalism).
>
> Why doesn't talk about sets commit one to platonism? Can you elaborate.
> Are sets not mathematical objects? Are not mathematical objects platonic
> objects? I was thinking that when one utters a sentence about sets, that
> she is either making a statement about platonic objects or her statement
> is a translation of a well-formed formula of set theory. And I would think
> that the desire to translate formal set-theory sentences into talk
> about objects indicates a platonic tendency.
[snip]
> I guess I must be confused about the definition of 'platonism.' Do some
> of these mathematicians believe that there is hope to produce substantial
> mathematics within some other formal framework? If not, doesn't that
> make them Platonists?
>
> I really am starting to believe that all mathematicians are platonists
> (and I don't intend that word to have a negative connotation). But
> of course, this belief could be wiped out, if my understanding of the
> word changes.
>
> Thanks for all your articulate comments,
> Leonard Blackburn
> Student, University of Minnesota
Here are some definitions from Borowski & Borwein, _Dictionary of
Mathematics_. HTH
Platonism, n. the philosophical theory that mathematical objects exist in
advance of and independently of our knowledge of them and of any physical
instantiation of them, and therefore that mathematical truth does not
consist in, but is the aim of, the construction of proofs.
[To oversimplify: that mathematical objects and truths are discovered, not
invented.]
realism, n. 1. the philosophical doctine that words refer to entities
that exist in reality, rather than merely being the signs of concepts or
sets of instances, whence in particular, that mathematical entities have a
real existence independent of our conception of them and of physical
representations or instances.
constructivism, n. The philosophical doctrine that mathematical entities
do not exist independently of our construction of them.
formalism, n. the philosophical doctrine that mathematical statements
have no extrinsic meaning but that their symbols themselves, regarded as
physical objects, exhibit a structure that has useful applications.
----------
IMO, working mathematicians tend to be Platonists, because the
mathematical entities they deal with certainly feel like they they have a
firm existence outside oneself. If you push on the entities, they push
back.
--
---------------------------
| BBB b \ barbara minus knox at iname stop com
| B B aa rrr b |
| BBB a a r bbb |
| B B a a r b b |
| BBB aa a r bbb |
-----------------------------
Paul Holbach
> news:<aa503d8.02092...@posting.google.com>...
> But aren't there only countably many logical predicates because
> there are only finitely many symbols in our language? Thus, this
> would seem to imply that there are only countably many sets.
> Wouldn't we like to believe that there are uncountably many sets?
> (Maybe I am misunderstanding, but in any case I still don't see how
> one can do without Platonism.)
Well, most mathematicians may "like to believe that there are
uncountably many sets", but the strength of wishful thinking is no
indication of truth...
If mathematical Platonism is the doctrine that mathematical entities
such as numbers, functions and sets are objectively real, existing
mind-independently just as the planet Pluto would be existing even if
nobody had ever discovered it, then Platonism is ridiculous nonsense.
Peirce writes that "a real thing is something whose characters are
independent of how any representation represents it to be". In this
sense mathematical entities are unreal, ie they are nothing but
semantic units. A semantic unit is a coherently interpreted theme
which is embedded in a complex semiotic network such as the system of
numbers. In this respect the question whether sets really exist is
absolutely pointless! Another question is how many sets there are.
Well, there are as many sets as there are specified sets. Of course,
the Platonist would instantaneously reply that specifiability was no
necessary criterion of the existence of mathematical objects such that
there were, for example, infinitely many unspecifiable and unnameable
numbers. But I doubt that eg "the set of all unspecifiable numbers" is
a consistent notion, for to me itæ„€ as indefensible as the talk of an
allegedly absolutely incomprehensible God. Discernibility and
existence are virtual synonyms! What cannot be denoted in principle is
nothing! The number of linguistic predicates is not actually, but
potentially infinite; so is the number of sets. Platonism in
mathematics is a comfortable habit sparing the mathematicians
intricate philosophical reflection that would just perplex them! ;-)
regards
PH
Paul Holbach
> news:<aa503d8.02092...@posting.google.com>...
> > Talk about sets does not commit one to platonism
> > (and the alternative is not extreme silly formalism).
> Why doesn't talk about sets commit one to platonism? Can you
> elaborate.
> Are sets not mathematical objects? Are not mathematical objects
> platonic
> objects? I was thinking that when one utters a sentence about sets,
> that
> she is either making a statement about platonic objects or her
> statement
> is a translation of a well-formed formula of set theory. And I would
> think
> that the desire to translate formal set-theory sentences into talk
> about objects indicates a platonic tendency.
Itæ„€ one of the most ineradicable and most pernicious philosophical
mistakes to assume that talking about something presupposes the real
existence of a particular object! Itæ„€ absolutely erroneous to read
"object" as "something mind-independently real"! There need not be
anything real which is a set in order for us to be able to make
meaningful statements containing "set"! The aboutness of signs doesnæ„’
imply any direct reference to reality whatsoever because signs are
correlated with other signs interpreting the former. Objects are not
mind-independently "out there", but intrinsic aspects of semiosis, ie
the cultural process of producing and interpreting signs. The word
"set" denotes the class of immanent objects <set>, but there is no
real thing that is a set. On the contrary, Itæ„€ semiosis itself that
creates objects, ie the objects of our mind. If a word such as "set"
is interpreted, ie defined, then there is a meaning of "set", viz
<set> such that "set" has become a theme. Sets are themes because the
word "set" possesses a determinate meaning. In other words, sets are
thematic objects of thought, they are the 'subject matter' of
potentially infinitely many propositions. The view that there had
already been sets even before man created the word "set" (or the
corresponding words in other languages) is sheer nonsense! And to
assume the existence of a third realm of objective Platonic ideas,
which is absolutely imperceptible in principle, is almost
psychopathological in my opinion! Platonism is a schizophrenic
ideology! Itæ„€ no accident that theology is Platonistic to the core!
Platonistic mathematics is formal theology... ;-)
regards
PH
Mitchell Smith
Leonard Blackburn wrote:
> Aatu Koskensilta <aatu.kos...@xortec.fi> wrote in message
>
> > This is the case with all mathematical objects. What is a function? What
> > is a natural number?
>
> Good point. But I was hoping that functions, numbers, and other
> objects
> are just sets. If that's not an aesthetic viewpoint, then at least I
> would still be able to do mathematics using just sets as the only
> objects.
>
The fact that mathematical objects have representations composed of sets does not
make them sets. It does, however, say something about the nature of sets.
>
> > There have been some attempts to derive the axioms of ZFC from certain
> > natural axioms pertaining the cumulative hierarchy. However, I don't
> > think you can ever have a "definition of set" in the sense you seem to
> > use. Sets are quite like functions. A function is an object for which
> > the application relation is defined and is "functional", nothing else
> > can be said about them. Similarly sets are objects for which the
> > membership relation is somehow given.
> >
Ah. The grand obfuscation. Compare topological model theory and
mereotopology--or, pointless topology--to discover that your problem is the use
of
the identity predicate as a 'logical' symbol of the language.
You can introduce a strict transitive predicate using a circularly referenced
definition. It has a self-defining syntax. With a strict transitive predicate,
you can introduce the membership predicate using a circularly referenced
definition.
The strict transitive predicate corresponds to the mereotopological interior part
predicate.
The use of circular reference constrains all extensional interpretations of these
predicates. Consequently, the object references introduced using these
predicates are not open to arbitrary interpretation as they are in conventional
model theory. That is, they have an intensional interpretation that is not
obfuscated by terms like "somehow" or "too big.".
The problem I am talking about here reflects on why the Generic Model Theorem
prevents anyone from accepting any extensional interpretation of ZFC as a model
of the class universe.
>
> > Even natural numbers have this property. What is a natural number? I
> > don't know of any good answer. However, we know what sorts of relations
> > natural numbers have with each other (and other objects).
> >
>
Spend some time reading Hume and Kant. Then, you will start to have some good
answers because you will understand the epistemic constraints associated with
object reference.
>
> > But what is a function? It's merely something that can be applied to
> > other things. The class of recursive functions is a subclass of the
> > class of all functions, and it's precisely what makes them different
> > from other functions (belonging to the closure of some function
> > formation operations, or being computable by Turing machine or being
> > lambda definable, &c) that you consider important. But this is merely
> > defining a *subclass* of the itself "undefined" (in the sense I infer
> > you use) class of functions.
>
A function is a language term containing free variables that has been
characterized with a well defined description. But, well definition has an
intrinsic dependency on a particular equivalence relation. How meaningful is
your function if you do not know the multiplicity of your equivalence classes?
That is, how can it be resolved to an object if it is referring to a plurality of
objects?
>
> Ok, I see what you mean. While we have a definition that identifies
> the recursive functions within the class of all functions, we don't
> have a definition of the class of all functions. So, my analogy was
> poor. In order to get a definition of a set somehow analogous to the
> definition of a recursive function, I would need a larger given
> context
> to work in.
>
You could, of course, ask yourself how to get the maximal class from the outset.
If you compare the axioms of group theory with those of ZFC, you notice a
striking difference. Every group satisfies the group axioms. ZFC is primarily a
listing of closure properties of the class universe. Sets do not satsify ZFC.
Why can't anyone else see that they are claiming that square blocks make round
holes?
Ask first what the axioms are describing and then ask what the term referents
have in common with that object. You will arrive at the conclusion that the
class universe is a connected topology for which every proper subspace is totally
disconnected.
>
> > > Or is talk about sets sometimes just abbreviated talk about well-formed
> > > formulas of set theory?
> >
> > Talk about sets does not commit one to platonism (and the alternative is
> > not extreme silly formalism).
>
I never saw a Venn diagram representation of a formula. However, logical forms
are topological in character. But, how do you sort it out?
>
> Why doesn't talk about sets commit one to platonism? Can you
> elaborate.
Are the common pigeonholes meaningful?
>
> Are sets not mathematical objects? Are not mathematical objects
> platonic
> objects? I was thinking that when one utters a sentence about sets,
> that
> she is either making a statement about platonic objects or her
> statement
> is a translation of a well-formed formula of set theory. And I would
> think
> that the desire to translate formal set-theory sentences into talk
> about objects indicates a platonic tendency.
>
Read Kant. In "Prolegomena to Any Future Metaphysics" he discusses the nature of
mathematical objects as a priori synthetic cognitions. In "Critique of Pure
Reason" he discusses the generation of a priori synthetic cognitions as the
general process whereby representations--either sensuous or imagined--are
instantiated as concepts.
Then try to make sense of the goat entrails of foundational mathematics.
>
> > > me for two reasons. (1) We don't know if ZFC has _any_ models at all.
> > > It might be an inconsistent theory.
> >
> > Well, in simlar vain it is possible that PA is inconsistent.
>
> Yes, of course.
>
Goedel's constructible universe is a model of ZF in which the axiom of choice
holds as a theorem.
Of course, its construction depends on whether or not the class of ordinals is a
meaningful concept. It is not a set. Reference to it depends on naive
metalinguistic usage of set-theoretic predicates.
>
> >
> > We have it in a sense backwards these days. Arguable for natural
> > numbers, and to a much lesser extent for sets, the original intention of
> > the axiomatisation was to characterise precisely (up to isomorphism)
> > what the structure we study is and what properties it haves by virtue of
> > being just that structure.
>
> But to think about the structure first is to be a platonist, isn't it?
>
Read Kant. Burn the goat entrails.
If you can figure him out, you will like him. His constructions have a binary
tree as their global structure.
>
> > Thus the structure comes first, and its
> > axiomatic description comes later. I find it rather odd that people go
> > about insisting that there is some real possibility that ZFC or PA is
> > inconsistent. They're not, they have models, namely natural numbers and
> > the cumulative hierarchy.
>
> But when I said that ZFC might be inconsistent I meant to make a very
> precise and true statement.
Then learn more classical set theory.
> Namely, using the rules of a sound and
> complete first-order predicate calculus (like the one I learned in
> Enderton's _A Mathematical Introduction to Logic_) it may be possible
> to derive from the axioms of ZFC a sentence and its negation. Are
> you saying this isn't possible?
Yes, provided you accept the construction strategy of Goedel's constructible
universe. Rejecting it places the burden of proposing a reasonable alternative
upon you, however.
> There is no proof within ZFC that
> ZFC is consistent, and the validity of any proof that ZFC is
> consistent in a larger theory would depend on the consistency of that
> larger theory.
The extensional characterization has nothing to do with deductions. There is a
whole chapter on models and satisfaction in Enderton. Too bad he is such a
terrible author.
>
> So, I don't know what you mean by saying that ZFC and PA have models.
> I grant you that it is hard to argue that our platonic structure of
> the
> natural numbers, about which we have much intuition, is not a model of
> PA.
> But I think our picture of the cumulative hierarchy of sets is much
> more hazy.
Yes. The seeds of doubt.
> At least it is for me. I haven't seen a clear, complete,
> and unambiguous definition of this hierarchy, even just in plain
> English.
>
What is so hard? It is a topological cover.
Of course, it depends on accepting reference to the ordinal numbers as
meaningful.
It also depends on the power set axiom. Unfortunately, mereotopology shows that
it is possible to interpret the order relation of the power set axiom
independently from the simple order relation of lax containment. Consequently,
the cumulative heirarchy has topological properties independent of the
extensional models which one might formulate for ZFC. In turn, this creates the
ambiguity of which Cohen's forcing takes advantage. Thus, you get the Generic
Model Theorem and the absolute inability to formulate an extensional
interpretation of the predicates of ZFC with respect to a class universe.
>
> Also, doesn't the belief that ZFC and PA are consistent rely, then,
> on the belief that these platonic structures exist and serve as
> models?
> And then, doesn't this mean that every mathematician is either a
> platonist
> or an extreme silly formalist who doesn't believe there is any meaning
> behind mathematics? (Maybe I don't comprehend the definition of
> 'platonist'.)
>
There are those goat entrails again.
>
> > (Of course, we could all be mistaken about
> > everything, but what is that supposed to mean in the first place?).
>
Computers interact across a network using network protocols.
How did we evolve to have effective communication? Suppose mathematics is the
"maximal portable communication grammar."
By grammar, I simply mean a body of rules. I prefer the term over that of
protocol.
By portable, I mean that any language user, through their own volition will
arrive at the same conclusions with regard to efficacy of communication
strategies.
By maximal, I mean to reflect severe epistemic constraints imposed by the fact
that different organisms have different subjective experiences.
By communication, I mean to reflect the detachment rules which allow us to accept
the conclusion of one deduction as the premise for another. This characterizes
the deductive calculus as a special form of salesmanship wherein a language user
who agrees with the premises proposed by another language user is compelled to
accept the conclusions and is permitted to adopt those conclusions as personal
convictions.
How does that compare with Pascal's statement that God gave us the natural
numbers?
ZFC is about object reference--not collections.
>
> > Your conception is certainly shared by most people who have superficial
> > and sometimes even non-superficial familarity with the incompleteness
> > results of logic. I personally favour the model theoretic view of world;
> > logic is the study of structures and associated languages, answering
> > such question as whether the language is rich enough to describe the
> > structure, what sort of properties leave a certain fragment, &c.
>
> Does that make you a platonist? I want to be clear on the definition.
>
No. You cannot be clear on the goat entrails philosophers burn on the alters.
>
> >
> > I find the view that logic is somehow the "foundation" of mathematics
> > rather odd; one needs quite an elaborate mathematical machinery to even
> > get to the interesting results of mathematical logic. Yet this view is
> > what underlies for example your conception of the role of logic. It is
> > perfectly reasonable from a normal mathematical point of view to use set
> > theory (as an informal theory of the cumulative hierarchy) to study the
> > metatheory of axiomatic set theory. From what other perspective should
> > we approach the study?
>
Language is topological. Logic is not the foundations of mathematics. Topology
is the foundation of logic.
>
> I don't know. But I feel the need that another perspective be
> invented.
>
I offered you that in another post and you did not even bother to ask one
question.
You could also look at mereotopology. However, they have the same problem with
naive metalinguistic usage of predicates.
>
> >
> > (As a side note, forcing and many other important results in axiomatic
> > set theory can be rendered into a form that perhaps pleases you more; it
> > is possible to view these as proof transformations. It is in general
> > possible, though not sensible, to interprete model theoretic result as
> > results about classes of languages and theories without reference to
> > models).
> >
>
> > > it is imagined that there is a model of ZFC, and an ordinary mathematical
> > > proof is given with that model in mind. Does this mean that all of these
> > > set theorists are Platonists?
> >
> > No, it means that they, as do virtually all mathematicians, realise that
> > there is no realistic hope to ever produce a mathematical work of
> > substance rendered in formal first order logic.
>
> I guess I must be confused about the definition of 'platonism.' Do
> some
> of these mathematicians believe that there is hope to produce
> substantial
> mathematics within some other formal framework? If not, doesn't that
> make them Platonists?
>
I do. I have. It is in your responses. Get curious. Lose the goat entrails.
>
> I really am starting to believe that all mathematicians are platonists
> (and I don't intend that word to have a negative connotation). But
> of course, this belief could be wiped out, if my understanding of the
> word changes.
>
Read Kant. Synthetic a priori cognitions.
>
> Thanks for all your articulate comments,
They were good--just conventional and leading to the same result as the last
century of set theory.
Good luck with your studies. Seriously, though. Look at mereotopology and
formal ontologies. It is getting very interesting with all that is going on in
knowledge management.
:-)
mitch
Douglas Eagleson
Leonard Blackburn wrote:
Well, one thing to remember is that a set recognized is not
the set of the theorist. And you defined the set observed
as the abstract set of logical theory and speculation.
So try to think of the relationship between a definition
and the thing to which it is applied. Except think of the
abstract definition and abstract thing. And of course the
theory of definition, that is, the abstract relation between
the two things in the previous sentence.
And the reason for the abstract set as a single thing is the
same as the reason for recognition of a real set.
And here this abstract set definition only applies those
things recognizable. So expand the definition to those
things recognized by any thing.
And the abstract recognition is the relation of set definition.
This is how I define a set, except recognition can be replace
by the term objective relation. Objective relation, meaning a
physically existent relation in nature, the sensory world.
Douglas Eagleson
Gaithersburg, MD USA
Aatu Koskensilta
Yes. And you're bound to have "undefined" primitive concepts within that
larger context. These concepts will have only "implicit" definitions, or
"functional" definitions if you wish.
>>>Or is talk about sets sometimes just abbreviated talk about well-formed
>>>formulas of set theory?
>>
>>Talk about sets does not commit one to platonism (and the alternative is
>>not extreme silly formalism).
>
>
> Why doesn't talk about sets commit one to platonism? Can you
> elaborate.
> Are sets not mathematical objects? Are not mathematical objects
> platonic
> objects? I was thinking that when one utters a sentence about sets,
> that
> she is either making a statement about platonic objects or her
> statement
> is a translation of a well-formed formula of set theory. And I would
> think
> that the desire to translate formal set-theory sentences into talk
> about objects indicates a platonic tendency.
Talk about sets as used in mathematical context does *not* commit one to
platonism. If this were the case there could be no non-platonistic
philosophy which were not highly restrictive to mathematical practice.
Some philosophies of mathematics do restrict mathematical practice,
namely finitism, intuitionism, constructivism and the like.
Platonism is the philosophy of mathematics which states that semantics
of mathematics is model theoretic, i.e. there is a model (the Platonic
realm of mathematical objects) which satisfies the propositions of
mathematics. Thus it is a philosophical model of semantics of the part
of our language that is used in mathematics.
However, merely speaking about sets certainly does not commit one to
platonism, to formalism or to any particular philosophical thesis about
semantics of mathematical language. Similarly, my speaking and having
learned a natural language does not commit me, say, Chomskyan thesis of
universal grammar.
Of course, platonism is a very suggestive and simple model for the
semantics of informal mathematics. It is not, I think, a very plausible
one, though.
>>>me for two reasons. (1) We don't know if ZFC has _any_ models at all.
>>>It might be an inconsistent theory.
>>
>>Well, in simlar vain it is possible that PA is inconsistent.
>
>
> Yes, of course.
>
>
>>We have it in a sense backwards these days. Arguable for natural
>>numbers, and to a much lesser extent for sets, the original intention of
>>the axiomatisation was to characterise precisely (up to isomorphism)
>>what the structure we study is and what properties it haves by virtue of
>>being just that structure.
>
>
> But to think about the structure first is to be a platonist, isn't it?
No. When you learned addition and multiplication, did you think of
formal systems? When you count things or figure out the properties of
the intersection of a particular set, do you think about formal systems?
Perhaps you do, but (most of the time) I do not.
>>Thus the structure comes first, and its
>>axiomatic description comes later. I find it rather odd that people go
>>about insisting that there is some real possibility that ZFC or PA is
>>inconsistent. They're not, they have models, namely natural numbers and
>>the cumulative hierarchy.
>
>
> But when I said that ZFC might be inconsistent I meant to make a very
> precise and true statement. Namely, using the rules of a sound and
> complete first-order predicate calculus (like the one I learned in
> Enderton's _A Mathematical Introduction to Logic_) it may be possible
> to derive from the axioms of ZFC a sentence and its negation. Are
> you saying this isn't possible? There is no proof within ZFC that
> ZFC is consistent, and the validity of any proof that ZFC is
> consistent in a larger theory would depend on the consistency of that
> larger theory.
This is true. The point I were trying to make was that the impossibility
of consistency proof in the theory itself is not really that
interesting. First of all, why would you trust the theory in the first
place? If you choose a weaker theory, why would you trust that one? The
only "trust" we can have in formal theories comes from our having in
mind a specific model in which the theory is satisfied.
> So, I don't know what you mean by saying that ZFC and PA have models.
> I grant you that it is hard to argue that our platonic structure of
> the
> natural numbers, about which we have much intuition, is not a model of
> PA.
We don't need the platonic structure, all we need are the natural
numbers of ordinary mathematics.
> But I think our picture of the cumulative hierarchy of sets is much
> more hazy. At least it is for me. I haven't seen a clear, complete,
> and unambiguous definition of this hierarchy, even just in plain
> English.
Here's one:
V(0) = {}
V(n+1) = V(n) U P(V(n))
V(a) = U of V(n) for all n < a, when a is a limit ordinal
Of course, this is not all that helpful.
> Also, doesn't the belief that ZFC and PA are consistent rely, then,
> on the belief that these platonic structures exist and serve as
> models?
What's the point of mathematics if the natural numbers don't "exist"?
> And then, doesn't this mean that every mathematician is either a
> platonist
> or an extreme silly formalist who doesn't believe there is any meaning
> behind mathematics? (Maybe I don't comprehend the definition of
> 'platonist'.)
By silly formalism I refer to the strange caricature of formalism
usually presented in the prefaces or introductory chapters of books on
foundations of mathematics; that mathematics is merely a game of symbols
and so forth.
The original formalism of Hilbert and co is bit more subtle and
sensible; certain propositions are held to be meaningful ("finitistic"
in some suitable sense of the word), whilst other are held to be
idealistic -- tools for producing results about these finitistic objects.
As an example, Gerog Kreisel has found a proposition p, of which the
shortest proof in FOL has more steps than the visible universe has
particles, but which has 5 step proof in second order logic. In this
case FOL proofs would be considered "finitistic" and second order proofs
"ideal".
>>(Of course, we could all be mistaken about
>>everything, but what is that supposed to mean in the first place?).
>
>
>>Your conception is certainly shared by most people who have superficial
>>and sometimes even non-superficial familarity with the incompleteness
>>results of logic. I personally favour the model theoretic view of world;
>>logic is the study of structures and associated languages, answering
>>such question as whether the language is rich enough to describe the
>>structure, what sort of properties leave a certain fragment, &c.
>
>
> Does that make you a platonist? I want to be clear on the definition.
No, it does not. Model theoretic view of logic is a *mathematical* view,
not philosophical. It's sort of a preference statement: we prefer to
look at logics from the point of view of their descriptive function,
instead of their deductive function. This view may or may not be
accompanied by certain philosophical views about the semantics of
mathematics. Mathematical realism might naturally lead to the model
theoretic view, but is neither necessary nor sufficient condition for it.
>>I find the view that logic is somehow the "foundation" of mathematics
>>rather odd; one needs quite an elaborate mathematical machinery to even
>>get to the interesting results of mathematical logic. Yet this view is
>>what underlies for example your conception of the role of logic. It is
>>perfectly reasonable from a normal mathematical point of view to use set
>>theory (as an informal theory of the cumulative hierarchy) to study the
>>metatheory of axiomatic set theory. From what other perspective should
>>we approach the study?
>
>
> I don't know. But I feel the need that another perspective be
> invented.
Why?
>>>it is imagined that there is a model of ZFC, and an ordinary mathematical
>>>proof is given with that model in mind. Does this mean that all of these
>>>set theorists are Platonists?
>>
>>No, it means that they, as do virtually all mathematicians, realise that
>>there is no realistic hope to ever produce a mathematical work of
>>substance rendered in formal first order logic.
>
>
> I guess I must be confused about the definition of 'platonism.' Do
> some
> of these mathematicians believe that there is hope to produce
> substantial
> mathematics within some other formal framework? If not, doesn't that
> make them Platonists?
More like intuitionists. It is not that mathematics can't be formalised;
of course any or even all the results of mathematics at a given point of
time *can* be formalised in some suitable framework.
For example, if you have strong enough computational framework (i.e. one
which sufficiently powrful concepts) for proving things which allows you
to use definitions, abbreviations and so forth, it is plausible that you
could reproduce most of the mathematics of today within, say,
Morse-Kelley set theory. Of course, as long as the system is understood
well enough we can diagonalise it and produce things which it can never
be used to prove. This alone does not show that all mathematics humans
will ever produce can't be formalised, merely that it can't be
formalised within a single tractable formalism.
> I really am starting to believe that all mathematicians are platonists
> (and I don't intend that word to have a negative connotation). But
> of course, this belief could be wiped out, if my understanding of the
> word changes.
I believe you're confusing philosophy of mathematics with the practice
of mathematics. Philosophy of mathematics is not determined by the
practice of mathematics (it is "underdetermined"). To some extent, the
practice of mathematics is determined by the philosophy of mathematics
one embraces.
John
Hi Aatu,
Perhaps you didn't catch my prior questions concerning a posting of
yours in the "What is a set" dialogue. So, here it is again.
> There have been some attempts to derive the axioms of ZFC from certain
> natural axioms pertaining the cumulative hierarchy. However, I don't
> think you can ever have a "definition of set" in the sense you seem to
> use. Sets are quite like functions. A function is an object for which
> the application relation is defined and is "functional", nothing else
> can be said about them. Similarly sets are objects for which the
> membership relation is somehow given.
What do you mean, SOMEHOW GIVEN? From the axioms governing "in", don't
the properties of this relation follow? And if so, when axioms
conflict--as (C3,C4) conflict with N4--isn't it so that one of these
of these is mistaken?
C3 EyAx[x in y <-> Et(x in t) & P(x)] (with y not free in P(x))
C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> x = y}]
N4 AxAy[Az(z in x <-> z in y) -> x = y]
That is, isn't the relation among (C4,N3,N4) as described by (1)?
1. FOL |- [((C3 & C4) -> ~N4) & (N4 -> ~(C3 & C4))]
--John
Bill Taylor
|> Borowski & Borwein, _Dictionary of Mathematics_.
Like most dictionaries, it is not so hot on fine shades of meaning.
|> Platonism, n. the philosophical theory that mathematical objects exist in
|> advance of and independently of our knowledge of them and of any physical
|> instantiation of them, and therefore that mathematical truth does not
|> consist in, but is the aim of, the construction of proofs.
THIS one is what I would have called "realism".
|> [To oversimplify: that mathematical objects and truths are discovered, not
|> invented.]
Not really oversimplified. Pretty accurate!
|> realism, n.
It's nice to see Platonism and realism distinguished, but the distinction
given is not all that helpful. Insofar as Platonism means anything at all
these days, it must refer to some absolute abstract existence to which our
everyday ideas of the realities of abstract objects only loosely conform.
I can't make much sense of this idea, & doubt if there are any modern proponents.
Most mathies would go for the top definition, and thus merely be realists.
|> 1. the philosophical doctine that words refer to entities
|> that exist in reality, rather than merely being the signs of concepts or
|> sets of instances, whence in particular, that mathematical entities have a
|> real existence independent of our conception of them and of physical
|> representations or instances.
This definition of B&B's is just a re-wroding of the top one. i.e. they don't
really distinguish between the two.
|> constructivism, n. The philosophical doctrine that mathematical entities
|> do not exist independently of our construction of them.
This is not really true, in modern usage. The definition given here is that
of *intuitionism*, which is a philosophical doctrine, whereas constructivism
is just a particular way of doing math. All intuitionists are constructivists,
but not vice versa. There are also constructivists who say "orthodox math
may be fine, but it gives a lot of conflated and unhelpful results, so we
are going to restrict ourselves to constructive methods".
Of course they realize that constructive math may ALSO have a lot of very
unhelpful results: e.g. every classical theorem P has a constructive
counterpart ~P v ~~P , but they would regard this metatheorem as being
essentially useless. So there is more than just dropping LET, though this
is central to its logical underpinnings.
|> formalism, n. the philosophical doctrine that mathematical statements
|> have no extrinsic meaning but that their symbols themselves, regarded as
|> physical objects, exhibit a structure that has useful applications.
IMHO this doctrine is actually *incoherent*. IOW self-contradictory.
They deny (or purport to) that all mathematical objects are not really
there and so do not have any properties; but they simultaneously insist
that propositional strings *are* there and *do* have certain properties.
So they deny that propositional strings of characters are legitimate math
objects, which is ridiculous. They admit their reality, but deny their
mathematicity, while simultaneously using mathlike logic to ascertain
their (real) properties!
This is clearly quite incoherent, and indeed ridiculous. I have no idea
what a formalist would reply to this crucial criticism, because I've never
met one. Can anyone else help answer this conundrum????
------------------------------------------------------------------------------
Bill Taylor W.Ta...@math.canterbury.ac.nz
------------------------------------------------------------------------------
The intuitionist conflates knowledge with truth;
The constructivist conflates infeasibility with impossibility.
------------------------------------------------------------------------------
Bill Taylor
|> The hope of making things "airtight" is a character trait that
|> tends to lead people to interest in the foundations of mathematics.
|> That was the way I felt too, long ago. I've come to the conclusion
|> that it's hopeless, and really not so desirable anyway.
This reflects my view also. I believe our proper aim is CLARITY, not certainty.
|> It's turtles all the way down -- and that's OK.
I don't really sympathise here, though. IMHO it is NOT turtles all the way
down. The bottom turtle is N itself, or PA itself (depending on whether you
want objects or statements to be basic). IMHO N with its PA is so utterly
crystal clear, that it "has no foundation, needs no foundation", to misquote
Boromir. It IS the foundation of everything else in math AND math logic.
All the technical work on "foundational logic" is really a lot less clear
than N/PA itself! And depends on it for meaning, anyway.
IMHO the rest of logic, exists to clarify the relationships of various more
complex elements of math to one another... independence results, reverse
mathematics, and so forth.
|> To get a mental concept of what a set is, I think the best thing
|> is to start small; let's talk about what a set of natural numbers is.
....
|> If that's all clear, then you need to visualize *all* such
|> sets being gathered into a completed whole, the powerset of the naturals.
....
|> There is one more piece: You have to make friends with
|> the notion of wellordering as an *informal* concept....
|> Then you abstract those lengths to get an informal notion of "ordinal".
This is a wonderful summary of the basis of modern mathematics, and its building
up from the integers. I wonder if I don't detect a smell of my own influence
in there Mike? - not that I actually want to claim any credit, but it reminds
me powerfully of this oft-used sig of mine...
------------------------------------------------------------------------------
Bill Taylor W.Ta...@math.canterbury.ac.nz
------------------------------------------------------------------------------
Set theory is a shotgun marriage - between well-ordering and power-set.
The two parties get along OK; but they hardly seem made for each other.
------------------------------------------------------------------------------
Bill Taylor
who writes:
|> There have been some attempts to derive the axioms of ZFC from certain
|> natural axioms pertaining the cumulative hierarchy.
True. I find the Dana Scott approach most congenial. A much underrated paper.
|> However, I don't
|> think you can ever have a "definition of set" in the sense you seem to use.
True. In effect, sets are defined by their use in math, or more basically
by their axiomatics in ZF(C), thought there are people (hello Torkel!) who
object very vociferously to this latter viewpoint.
|> Talk about sets does not commit one to platonism
Perhaps; but as Mike Oliver often notes, working with them *does* so commit one,
at least temporarily. And it's hard for most of us to work with sets but
simultaneously merely "pretend" that they exist.
|> > (1) We don't know if ZFC has _any_ models at all.
|> > It might be an inconsistent theory.
|> Well, in simlar vain it is possible that PA is inconsistent.
Sternly resisting the temptation to do a spelling flame, (and indeed it is
such a serendipitous one it may well be intended!), I nonetheless note that
this is not really a fair comparison.
To conceive of the possibility of PA being inconsistent is fair game for
ridicule, IMHO. But to conceive of an inconsistency in ZF is not nearly
so heinous; after all the alleged model is *nowhere* near as viewable and
verifiable as is N for PA. I myself do not for a moment expect ZF ever to
turn out to be inconsistent, but it is not a laughable idea. We do not
really grok V all that well! (OK so I speak for myself, but I can maintain
a reasonable suspicion of those who blithely claim otherwise.)
|> We have it in a sense backwards these days. Arguable for natural
|> numbers, and to a much lesser extent for sets, the original intention of
|> the axiomatisation was to characterise precisely (up to isomorphism)
|> what the structure we study is and what properties it haves by virtue of
|> being just that structure.
I do agree with all that, however.
(I might interject "alleged" structure for V, but only when being bloody-minded.)
|> > It seems illogical to interpret the theorems of set theory
|> > (ZFC) in a set (equipped with a binary relation, which is itself a set)
|> > itself. Maybe not illogical, but certainly strange.
This earlier complaint is often made, and is a fair one. I think it is
simply answered by noting that the amount of set theory required for
a meta-theory of the models for a logic is FAAAAAAaaaaar far less than
the amount given to us by ZF and its V. The set-iness you have to swallow
for merely modelling FOL is quite digestible.
|> I find the view that logic is somehow the "foundation" of mathematics
|> rather odd; one needs quite an elaborate mathematical machinery to even
|> get to the interesting results of mathematical logic.
Exactly so; as in my parallel post. N founds logic, not the other way round!
------------------------------------------------------------------------------
Bill Taylor W.Ta...@math.canterbury.ac.nz
------------------------------------------------------------------------------
a priori a posteriori
.--------------------------.
synthetic | math | science |
|------------+-------------| The KANT-TAYLOR bi-dichotomy.
analytic | logic | sophistry |
`--------------------------'
Mike Oliver
>
> I don't really sympathise here, though. IMHO it is NOT turtles all the way
> down. The bottom turtle is N itself, or PA itself (depending on whether you
> want objects or statements to be basic). IMHO N with its PA is so utterly
> crystal clear, that it "has no foundation, needs no foundation", to misquote
> Boromir. It IS the foundation of everything else in math AND math logic.
> All the technical work on "foundational logic" is really a lot less clear
> than N/PA itself! And depends on it for meaning, anyway.
There's not too much point in rehashing this as we've been over it so
many times. As you know I don't see it that way; I view the (arbitrarily
large) natural numbers as an abstraction of much the same kind that
the wellfounded sets are. I do not see any sharp lines between the
completely abstract and the completely concrete; the only demarcation
of the sort I can admit is, "if the existence of such-and-such were refuted,
would I have a rational response?" Clearly there *would* be a rational
response if arbitrarily large naturals were refuted, and no, it
would not be jumping out the window.
I keep looking for a good opening to expound my thoughts in detail
on monotonic versus nonmonotonic knowledge; this might be it, except
that I don't feel like it at the moment. Let me just say that
because of the above considerations, I put our knowledge about
the natural numbers in the nonmonotonic category.
>|> There is one more piece: You have to make friends with
>|> the notion of wellordering as an *informal* concept....
>|> Then you abstract those lengths to get an informal notion of "ordinal".
>
> This is a wonderful summary of the basis of modern mathematics, and its building
> up from the integers. I wonder if I don't detect a smell of my own influence
> in there Mike? - not that I actually want to claim any credit, but it reminds
> me powerfully of this oft-used sig of mine...
Actually my epiphany vis-a-vis informal ordinals came from reading
the "Birthday Cantatatata" in _Goedel_Escher_Bach_, *long* after
I was already quite familiar with their properties. However there
is no doubt that you have been a part of the process as I
refine my foundational views.
Bill Taylor
|> The fact that mathematical objects have representations composed of sets does
|> not make them sets. It does, however, say something about the nature of sets.
Hey! Very nice summary of the situation!
And what it says about sets is why we love and cherish them so well.
They were amazingly, unexpectedly, and unutterably CONVENIENT as a grand
unifying device for all previously semi-disconnected branches of math.
And, modulo complaints from categorists, have continued to be so and look
to remain so for the forseeable future.
|> Compare topological model theory and mereotopology--or, pointless topology--
OUCH! That was an unfortunate near-compelling mistaken-uptake!
But yes, point-free topology does sound like a plausible good idea.
But Mitch - if you want a lot of people to understand that it ISN'T actually
*pointless*, you had better start off much more simply than you have been doing
so far. Just regard us as all rather dumb but vacuously willing to consider
a new idea if we're led to it by the nose with firm hand-holding!
Why don't you write a briefer introductory article, outlining the basic
ideas of point-free topology, in INFORMAL outline only, with little or no
technicalities, and how it might apply to logic; again without technicality.
After we have grokked what it might be about, we will then be in a position
to ask for particular technical details, so maybe make some constructive crticism.
Remember Einstein said we ought to be able to outline our theories to our
grannies and charladies!
,---,_ ,
_> `'-. .--'/
.--'` ._ `/ <_
>,-' ._'.. ..__ . ' '-.
.-' .'` `'. '.
> / >`-. .-'< \ , '._\
/ ; '-._> <_.-' ; '._>
`> ,/ /___\ /___\ \_ /
`.-|(| \o_/ \o_/ |)|`
\; \ ;/
\ .-, )-. /
/` .'-'. `\
;_.-`.___.'-.;
------------------------------------------------------------------------------
Bill Taylor W.Ta...@math.canterbury.ac.nz
------------------------------------------------------------------------------
From its own POV light takes no time at all to get from source to sink,
the two events are simultaneous. So photons *cannot* "wear out".
------------------------------------------------------------------------------
R. Srinivasan
>
> IMO, working mathematicians tend to be Platonists, because the
> mathematical entities they deal with certainly feel like they they have a
> firm existence outside oneself. If you push on the entities, they push
> back.
Here is a link to Prof. Stephen Simpson's opinion on the philosophy of
mathematics:
http://www.math.psu.edu/simpson/papers/philmath/node20.html
I quote the final paragraph of this reference:
"We have mentioned three competing 20th century doctrines: formalism,
constructivism, set-theoretical Platonism. None of these doctrines are
philosophically satisfactory, and they do not provide much guidance
for mathematically oriented scientists and other users of mathematics.
As a result, late 20th century mathematicians have developed a split
view, a kind of Kantian schizophrenia, which is usually described as
``Platonism on weekdays, formalism on weekends''. In other words, they
accept the existence of infinite sets as a working hypothesis in their
mathematical research, but when it comes to philosophical speculation,
they retreat to a formalist stance. Thus they have given up hope of an
integrated view which accounts for both mathematical knowledge and the
applicability of mathematics to physical reality. In this respect, the
philosophy of mathematics is in a sorry state."
Sincerely,
R. Srinivasan srad...@in.ibm.com
Daryl McCullough
>Actually my epiphany vis-a-vis informal ordinals came from reading
>the "Birthday Cantatatata" in _Goedel_Escher_Bach_, *long* after
>I was already quite familiar with their properties.
That's nice to hear! When _Goedel, Escher, Bach_ first came out, my
professional logician friends were scathing in their criticism. They
considered it a "Gee-Whiz" book that needlessly played up the
mysterious and paradoxical side of its subject matter, without
conveying any actual skills or knowledge to the layman reader. These
people were thoroughly sick of the hype given to the driving idea of
the book, self-reference.
I completely disagreed at the time. I thought that it was a blast
to read, and it showed me the joy of math and logic in a way that
a textbook almost never can. In contrast to many pop books on the
subject of Goedel, Hofstaeder's book actually gave the details for
understanding how Goedel's theorm could be proved.
I took my disagreement with the logicians as evidence that I
wasn't cut out to be a logician, but it's nice to know, Mike,
that some real mathematicians got something out of it, as well.
--
Daryl McCullough
Ithaca, NY
Daryl McCullough
>I don't really sympathise here, though. IMHO it is NOT turtles all the way
>down. The bottom turtle is N itself, or PA itself (depending on whether you
>want objects or statements to be basic). IMHO N with its PA is so utterly
>crystal clear, that it "has no foundation, needs no foundation", to misquote
>Boromir. It IS the foundation of everything else in math AND math logic.
>All the technical work on "foundational logic" is really a lot less clear
>than N/PA itself! And depends on it for meaning, anyway.
Do you mean Boromir the character from _The Lord of the Rings_?
Actually, you are right that the natural numbers can be considered
the foundation of all of mathematics. However, the aspect of number
theory that is most important for modern mathematics is the idea
of natural induction. But the principle of natural induction is
really about *collections* of numbers, rather than numbers. So I
think that there is a tiny bit of set theory present in even the
most elementary number theory.
Leonard Blackburn
> Here are some definitions from Borowski & Borwein, _Dictionary of
> Mathematics_. HTH
>
> Platonism, n. the philosophical theory that mathematical objects exist in
> advance of and independently of our knowledge of them and of any physical
> instantiation of them, and therefore that mathematical truth does not
> consist in, but is the aim of, the construction of proofs.
>
> [To oversimplify: that mathematical objects and truths are discovered, not
> invented.]
>
> realism, n. 1. the philosophical doctine that words refer to entities
> that exist in reality, rather than merely being the signs of concepts or
> sets of instances, whence in particular, that mathematical entities have a
> real existence independent of our conception of them and of physical
> representations or instances.
>
> constructivism, n. The philosophical doctrine that mathematical entities
> do not exist independently of our construction of them.
>
> formalism, n. the philosophical doctrine that mathematical statements
> have no extrinsic meaning but that their symbols themselves, regarded as
> physical objects, exhibit a structure that has useful applications.
>
> ----------
>
> IMO, working mathematicians tend to be Platonists, because the
> mathematical entities they deal with certainly feel like they they have a
> firm existence outside oneself. If you push on the entities, they push
> back.
Thanks for the definitions. I sm still confused, however--particularly with
the word "exist". Maybe you or others could help me with the following
scenarios. Which of the following is a platonist belief? I use "I" below
even though I might not believe all of these things:
1. I believe that N (the set of natural numbers) "exists" as a completely
coherent idea and that <N,0,s> is a model of the Peano axioms. [notes: I put
"exists" in quotation marks because I haven't defined this word here. I
certainly don't mean that N physically exists like a chair does.]
2. I believe that the Goldbach conjecture (that every even natural number
> 2 is the sum of two primes) is a meaningful statement that has a definite
truth value (even if we may never discover this truth value and even if the
Goldbach conjecture turns out to be independent of ZFC).
3. I believe that there exists a coherent notion of "set," that the
cumulative type hierarchy of sets (V) is a coherent idea that mathematicians
have begun to describe (even if the description isn't complete), and that
the axioms of ZFC are true statements about V.
I may have more to add to the list later. Feel free to add any others
that may help in clarifying the definition of "platonist" for me.
I have spoken with other mathematicians (students) who believe that
the word "platonist" has a negative connotation, and that it is ridiculous
to believe that mathematical objects somehow exist as ideal things in
some Platonic realm, and that it is also ridiculous to believe that
mathematical objects exist even before humans have "discovered" them.
They have likened this to faith or religion. (Just a couple of years
ago, I was one of these people who also went so far as to think that
"mathematical truth" is not a coherent concept and has no place in
mathematics.)
I think this derision of platonism comes from a disagreement on
the definition of "exist". When I think of the integers as existing, I
probably do not mean the same thing that Plato himself did. I do not
imply some sort of physical or spiritual existence, or that there is some
"location" where such objects might exist. Although I cannot adequately
explain what I mean by "exist", it is a much weaker concept than that
just described.
Also, with regard to believing that the integers and other mathematical
objects existed before humans, and that humans discovered them, I can think
of more than one point of view that hinge on the definition of the word
"exist". For example, did the constellations (Ursa Major, etc.) exist
before humans named them? Well, the stars were there (and in close to their
same positions) before we named the constellations. The different
configurations of stars were there, visible. So in a sense they did exist.
But one can equally argue that they did not exist, since a "constellation"
is more than just the stars themselves, it is a human "idea". Similarly,
did the number
1348172049856149287510329841732489612395786104976510947510
1248517431248510340494908948598455757751283475109843484112
exist before I just wrote it? I doubt that any other human being has
ever written this number before, or has even looked upon it before.
But one could argue it existed years ago anyway. Is this a belief
worthy of derision?
If (1)-(3) above are platonic beliefs, then how can any working mathematician
claim to be a non-platonist? If you don't believe in the universe of
sets, then how can you justify writing down the axioms of ZFC? The axioms
and statements of mathematics seem to be motivated by platonist beliefs.
But I might be wrong on this. I am still quite inexperienced (I read my
very first logic book just 2.5 years ago). Also, I'm not sure of the
difference between realism and platonism, and I will have to peruse the
comments given in the message below. It seems from the definitions above
that every platonist is a realist but not conversely. Both seem to believe
in the existence of mathematical objects. The difference seems
to be whether or not you believe that these various mathematical objects
existed _before_ anyone thought of them.
Josh Purinton
Leonard Blackburn <blac...@math.umn.edu> wrote:
>Does anyone know where I can read more about exactly what sets are?
In _from Boolean Algebra to Unified Algebra_,
(http://www.cs.toronto.edu/~hehner/BAUA.pdf) E. C. R. Hehner talks
about presenting sets via 'bunches':
In the notation commonly used for small sets, such as {1, 3, 7},
the comma was introduced as just punctuation, not as a mathematical
operator. As soon as the notation is introduced, we must say that
the order in which elements are written is irrelevant so that {1,2}
= {2,1}; the way to say that formally is A,B = B,A (comma is
commutative). We must also say that repetitions of elements are
irrelevant so that {3,3} = {3}; the way to say that formally is
A,A = A (comma is idempotent). And we should say that comma is
associative A,(B,C) = (A,B),C so that parentheses are unnecessary.
Evidently the comma can be seen as a mathematical operator with
algebraic properties that aggregates elements into a structure that
is simpler, more primitive, than sets; let us call them bunches.
Even the curly braces can be seen as an operator that applies to
a bunch and makes a set; its inverse ~ applies to a set and makes
a bunch: ~{1,2} = 1,2.
When a child first learns about sets, there is often an initial
hurdle: that a set with one element is not the same as the element.
It would be easier to present a set as packaging: a package with
an apple in it is obviously not the same as the apple. Just as {1}
and 1 differ, so {1,2} and 1,2 differ. Bunch theory tells us about
aggregation; set theory tells us about packaging. The two are
independent.
[...] Formalization of the lowly comma leads to a beautiful and
useful algebra.
Hehner also wrote _Unified Algebra_, a companion paper with more
technical details (http://www.cs.toronto.edu/~hehner/UA.pdf).
Josh
--
Josh Purinton <jo...@joshpurinton.com>
I have always thought the actions of men the best interpreters of their
thoughts. - John Locke
--
Josh Purinton <jo...@joshpurinton.com>
I have always thought the actions of men the best interpreters of their
thoughts. - John Locke
John
> (Barb Knox) writes a nice summary from Borowski/Borwein:
>
> |> Borowski & Borwein, _Dictionary of Mathematics_.
>
> Like most dictionaries, it is not so hot on fine shades of meaning.
>
> |> Platonism, n. the philosophical theory that mathematical objects exist in
> |> advance of and independently of our knowledge of them and of any physical
> |> instantiation of them, and therefore that mathematical truth does not
> |> consist in, but is the aim of, the construction of proofs.
>
> THIS one is what I would have called "realism".
>
> |> [To oversimplify: that mathematical objects and truths are discovered, not
> |> invented.]
>
> Not really oversimplified. Pretty accurate!
>
In view of (1),
1. FOL |- [((C3 & C4) -> ~N4) & (N4 -> ~(C3 & C4))]
C3 EyAx[x in y <-> Et(x in t) & P(x)] (with y not free in P(x))
C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> x = y}]
N4 AxAy[Az(z in x <-> z in y) -> x = y]
if you are a Platonist as per the above, you have no choice--or so it
would seem--but to grant that (C3,C4) apply to the relata of *in* and
*=* only if N4 does not; and (B) that N4 applies to the relata of *in*
and *=* only if either C3 or C4 does not.
Am I mistaken about this?
--John
Barb Knox
blac...@math.umn.edu (Leonard Blackburn) wrote:
Yes, that is the nub of the issue. Philosophers have been wrestling with
that for a very long time (at least since Plato).
> Maybe you or others could help me with the following
> scenarios. Which of the following is a platonist belief? I use "I" below
> even though I might not believe all of these things:
>
> 1. I believe that N (the set of natural numbers) "exists" as a completely
> coherent idea and that <N,0,s> is a model of the Peano axioms. [notes: I put
> "exists" in quotation marks because I haven't defined this word here. I
> certainly don't mean that N physically exists like a chair does.]
This example opens up a lot of issues. Whether or not something "exists
as [an] ... idea" can be viewed as a matter of psychology (for an
individual's ideas) or sociology (for the collected ideas of a society).
Since people and societies can and do have delusions, that sort of
existence is pretty weak.
Whether or not a particular idea is "completely coherent" is a matter of
logic. Ideas that are internally contradictory have a very weak claim
indeed to any sort of existence.
A modern mathematical Platonist would consider that some mathematical
entity "exists" iff it has a model. This of course shoves the issue from
the "existence" of the entity to the "existence" of the model.
Plato himself would argue that a material object such as your chair only
"exists" in the material universe by virtue of it being a reflection of
"the ideal chair" existing in the universe of forms. In this view, the
natural numbers actually have a more solid claim to "existence" than
chairs do!
And note that even if one believes that each finite natural number
"exists", it is a large philosphical leap to then infer the "existence" of
a single infinite entity which is the collection of all the natural
numbers at once.
> 2. I believe that the Goldbach conjecture (that every even natural number
> > 2 is the sum of two primes) is a meaningful statement that has a definite
> truth value (even if we may never discover this truth value and even if the
> Goldbach conjecture turns out to be independent of ZFC).
I don't understand this. Does the rhetorical "I" believe that (e.g.) the
Axiom of Choice has a definite truth value in ZF even though it has been
proven to be independent? If so, which of "true" or "false" is it?
> 3. I believe that there exists a coherent notion of "set," that the
> cumulative type hierarchy of sets (V) is a coherent idea that mathematicians
> have begun to describe (even if the description isn't complete), and that
> the axioms of ZFC are true statements about V.
A hard-core formalist could believe that just as sincerely as a
Platonist. Saying that "there exists a coherent notion of ..." is
essentially a matter of logic, which can be dealt with by purely formal
manipulations.
> I may have more to add to the list later. Feel free to add any others
> that may help in clarifying the definition of "platonist" for me.
>
> I have spoken with other mathematicians (students) who believe that
> the word "platonist" has a negative connotation,
Try speaking with some professional mathematicians about this. Like I
said above, working mathematicians tend towards Platonism.
> and that it is ridiculous
> to believe that mathematical objects somehow exist as ideal things in
> some Platonic realm,
Lots of things that seem "ridiculous" happen to be matters of experimental
fact (e.g., quantum mechanics, relativity, the fact that the earth moves
around the sun).
> and that it is also ridiculous to believe that
> mathematical objects exist even before humans have "discovered" them.
The notion that "God is a mathematician" gained currency as more and more
of the measurable physical universe was seen to have deep underlying
mathematical structure. So the notion that mathematics is in some way
prior to the physical universe is not ridiculous on the face of it.
> They have likened this to faith or religion. (Just a couple of years
> ago, I was one of these people who also went so far as to think that
> "mathematical truth" is not a coherent concept and has no place in
> mathematics.)
A little philosophy is a dangerous thing...
[snip]
Franz Fritsche
>>
>> I believe that the Goldbach conjecture (that every even natural number
>> > 2 is the sum of two primes) is a meaningful statement that has a definite
>> truth value (even if we may never discover this truth value and even if the
>> Goldbach conjecture turns out to be independent of ZFC).
>>
In fact, if it turns out that the Goldbach conjecture is independent
from ZFC it must be true! :-)
[ Since in this case there couldn't be a counterexample, that would
render the Goldbach conjecture false. ]
F.
Paul Holbach
> Thanks for the definitions. I sm still confused, however--particularly with
> the word "exist".
Hereæ„€ an important text (well, almost everything he wrote is
important...) by Peirce:
(taken from "Writings of Charles Sanders Peirce" Vol. 2/9.,p.103f:
see "The Peirce Edition Project", http://www.iupui.edu/~peirce/ )
"Logic must begin with analyzing the meanings of certain words, which
we shall take up in due order.
The first of these is the word "is", as when we say, Julius Caesar is
dead, a griffin is a fabulous animal, a four-sided triangle is an
absurdity, height is the distance from the ground, nothing is that
which does not exist. These examples suffice to show that we apply
this word to whatever we give a name, whether it really exists or not,
or whether we consider it as existing or not.
The word is is called by logicians the copula because it joins subject
and predicate. That which is, in the sense of the copula, was termed
ens (pl: entia) by the schoolmen, and the corresponding abstract noun
used was entitas. In this as in many other cases, we have taken in
English, the abstract noun in a concrete sense, and we can
consequently speak of entities. At the same time we have forgotten the
very general meaning attached to the word in the middle ages, as
denoting whatever can be named, and employ it for what would then have
been termed ens reale. Thus, we often hear the schoolmen reviled
because they considered abstractions to be "entities," but in their
sense of the term it admits of no dispute that an abstraction is ens.
It is true that they frequently use the word ens simply when they mean
ens reale, but only in cases in which there can be no doubt of their
meaning; and it was universal to consider entia as embracing not only
entia realia but also entia rationis. I propose to restore the term
ens or entity to its original meaning of whatever can be named or
talked about. I shall also endeavor as much as possible to reserve the
word being and other derivatives of is, to express this same
conception; but these words must be somewhat ambiguous.
It may be observed that entity is so extremely general a name that it
has no negative over against it. We may talk of a nonentity, but then
as we have given it a name it is also an entity.
In contrast with this general being which is conferred by our mere
thought of an object, is the being of real things which is quite
independent of what we think.
We shall designate this by "reality", and its cognates; and shall
employ "figment" and "fiction" to denote that which is non-existent
without meaning to imply that the conception has been a deliberate
invention.
It is important to observe that the essential difference between a
reality and a nonreality, is that the former has an existence entirely
independent of what you or I or any number of men may think about it.
What I dream, for example, only exists so far as my dreaming
imagination creates it. But the fact that I have had such a dream,
remains true whether I ever reflect upon that fact or not. The dream,
therefore, as a mental phenomenon, is a reality; but the thing dreamed
is a figment. If there ever really was such a man as Romulus, he would
have existed just the same if history had never mentioned him; but if
he is not a reality he exists only in the fables which have been told
of the foundation of Rome. When Gray says,
Full many a gem of purest ray serene
The dark unfathomed caves of ocean bear;
Full many a flower is born to blush unseen
And waste its sweetness on the desert air;
he expresses with precision the essential character of reality. But
when we say that the real is that which is independent of how you or I
or any number of men think about it, we have still left the conception
of independent being to be analyzed. Before making that analysis we
must consider the conceptions of one, two, and three.
We have seen that an ens is something to which the copula is can be
applied. But is is a word whose meaning is not complete in itself. It
means nothing to say that anything is (in the sense of the copula)
unless I say what it is; for the only function of the copula is to
join subject and predicate. Hence, whatever is, is somehow. This
somehow of entity I propose to express by the term quality. A quality
therefore in the very general sense in which I shall use it, denotes
whatever can be expressed by all that comes after is in a complete
assertion. Every ens, then, has some quality for to say that it is an
ens is to say that it may be made the subject of an assertion and that
assertion must have some predicate. There is no conception so vague
that some thing cannot be asserted of the object of it, for it is the
first condition of thought that some quality must be thought in the
thought."
The two most relevant statements of this text are the following ones:
a) I PROPOSE TO RESTORE THE TERM ENS OR ENTITY TO ITS ORIGINAL MEANING
OF WHATEVER CAN BE NAMED OR TALKED ABOUT.
b) EVERY ENS, THEN, HAS SOME QUALITY FOR TO SAY THAT IT IS AN ENS IS
TO SAY THAT IT MAY BE MADE THE SUBJECT OF AN ASSERTION AND THAT
ASSERTION MUST HAVE SOME PREDICATE.
So an entity "is born" by dint of an arbitrary noun being made the
subject of an assertoric sentence! Entities are interpreted nouns,
Being is immanent in language and being an entity means being the
subject matter of a particular discourse! And being the theme of a
text or a discourse certainly doesnæ„’ entail the real existence of
what is being talked about!
For example, there is an entity named '4' because there are semiotic
statements such as '4 is an even natural number.'. There would be no
number 4 if the sign '4' had never been made the subject of some
proposition!
regards
PH
Mitchell Smith
Bill Taylor wrote:
> Mitchell Smith <mit...@enteract.com> writes:
>
> |> The fact that mathematical objects have representations composed of sets does
> |> not make them sets. It does, however, say something about the nature of sets.
>
> Hey! Very nice summary of the situation!
>
> And what it says about sets is why we love and cherish them so well.
> They were amazingly, unexpectedly, and unutterably CONVENIENT as a grand
> unifying device for all previously semi-disconnected branches of math.
>
> And, modulo complaints from categorists, have continued to be so and look
> to remain so for the forseeable future.
>
> |> Compare topological model theory and mereotopology--or, pointless topology--
>
> OUCH! That was an unfortunate near-compelling mistaken-uptake!
> But yes, point-free topology does sound like a plausible good idea.
>
> But Mitch - if you want a lot of people to understand that it ISN'T actually
> *pointless*, you had better start off much more simply than you have been doing
> so far. Just regard us as all rather dumb but vacuously willing to consider
> a new idea if we're led to it by the nose with firm hand-holding!
>
> Why don't you write a briefer introductory article, outlining the basic
> ideas of point-free topology, in INFORMAL outline only, with little or no
> technicalities, and how it might apply to logic; again without technicality.
>
First, let me offer a simple apology without explanation. It is deserved.
Second, I have spent twenty years pursuing intuitions rejecting certain assumptions
governing the use of formal systems. My results are reasonable but incredibly
convoluted.
In simplest terms, the foundations of mathematics should be based upon defined
language elements. Formal systems using extensional interpretations of undefined
language primitives have their uses. But, they fail to generate model specification
for the class universe.
I infer from all of the literature that everyone believes Russell's paradox to be a
condition of reality. I view it as a reflection of illiteracy. The paradox itself is
not subject to challenge. The presumption that it precludes an intensional
interpretation of the universal quantifier as the set of all sets is the issue.
If someone will ask questions in small pieces, I will gladly explain this perspective
in small pieces. But, I cannot make it easy.
As for pointless topology...
Pointless topology, which I shall hereafter refer to as mereotopology, is not a
solution. But, it does justify a decision I made about fourteen years ago. You see,
when I started looking for definitions to construct my language, I had to come to
terms with a strict transitive order that most people simply take as being derived
from the membership predicate.
The definition for "proper_part" is given by:
for all a for all b( a proper_part b iff (
forall c( b proper_part c implies a proper_part c )
and
exists c( a proper_part c and not (b proper_part c) ) ) )
If you write this out on paper, just use the 'proper_subset' symbol. As noted, it
is a strict transitive predicate.
My second language symbol is the membership relation.
The definition for "in" is given by:
for all a for all b( a in b iff (
for all c( b proper_part c implies a in c )
and
exists c( a in c and not (b proper_part c) ) ) )
You see, I can formally define the membership predicate if I have a strict
transitive predicate. But, the strict transitive predicate comes first and has to be
subject to interpretation independent of interpretation for the membership predicate.
It took me two years to be comfortable with this idea. This year, I discovered a
formal system called mereotopology whose topological predicate is a strict transitive
relation. So, my decision concerning the viability of independent interpretation
reflects similar decisions by others.
But, my objective still lies with set theory. So, while I begin with a mereological
predicate, it will eventually be reinterpreted as strict containment through axioms
and theorems.
No more lest I find myself having to offer another open apology.
Someone should write out the formal sentences and ask why the syntax is parallel.
Thank you for the criticism.
:-)
mitch
Bill Taylor
|> By silly formalism I refer to the strange caricature of formalism
|> usually presented in the prefaces or introductory chapters of books on
|> foundations of mathematics; that mathematics is merely a game of symbols
|> and so forth.
This is indeed silly, but I wonder whether in fact, has any mathematician
actually *espoused* this view? I tend to doubt it. It strikes me as being
an (as usual) only partly understood and furtherly dumbed-down version by
various popularisers. And I think vital parts have been left out, though
I can't say exactly what, not being all that au fait with "official" formalism.
|> The original formalism of Hilbert and co is bit more subtle and
|> sensible; certain propositions are held to be meaningful ...
|> ... whilst other are held to be idealistic -
|> - tools for producing results about these finitistic objects.
This actually strikes me as being a remarkably sound and defensible view.
AFAICUI it seems to say: yes the naturals are *really* there, in ontological
space, and some things are definitely true about them, and that PA is among these.
And more precisely, it gives a very hard-edged criterion for the "meaningfulness"
of arithmetical (number-theoretic) propositions. In that when P is a decidable
(i.e. recursive) property of naturals, & "n" a numeral, then statements of form
P("n") are fully meaningful - can be tested either way;
Ax P(x) are falsifiable-if-false, (but not verifiable), I dub them "scientific";
Ex P(x) are verifiable-if-true, (but not falsifiable), I dub these "religious";
AE or EA are neither falsifiable nor verifiable; and thus actually meaningless
in themselves, but acquire meaning when used in logical conjunction
with the earlier sort, especially the first type.
It's not unlike the situation in science whereby there are many anti-positivistic
concepts still held to be meaningful because of their untanglable relation to
mere "observation statements". So in science the positivists were silly;
the more inclusive view of e.g. Deutsch, which includes "explanations" as
almost coeval with observations, is more sensible. Similarly a strictly
finitist view of math, that only decidable propositions are meaningful,
(?"strict finitism"?), is pretty silly. It should at least extend to
statements of the 2nd & 3rd types, which are semi-decidable, and ultimately
(a la Hilbert) to the 4th, by considering "internal consistency" and their
"untanglability" from the rest.
So basically I'm with Hilbert on this.
And I don't, and I don't imagine that Hilbert did, extend this to ZFC with
full confidence - certainly not to ZFC + LCs. And I'm not all *that* sure
about impredicative analysis, come to that.
So I think we could be called "limited realists" - we're committed to the
realism of N (and its equivalent minor extensions), but increasingly more
agnostic the further away from it one gets.
------------------------------------------------------------------------------
Bill Taylor W.Ta...@math.canterbury.ac.nz
------------------------------------------------------------------------------
Science - believing what you see
Religion - seeing what you believe
------------------------------------------------------------------------------
R. Srinivasan
>
> Thanks for the definitions. I sm still confused, however--particularly with
> the word "exist". Maybe you or others could help me with the following
> scenarios. Which of the following is a platonist belief? I use "I" below
> even though I might not believe all of these things:
>
> 1. I believe that N (the set of natural numbers) "exists" as a completely
> coherent idea and that <N,0,s> is a model of the Peano axioms. [notes: I put
> "exists" in quotation marks because I haven't defined this word here. I
> certainly don't mean that N physically exists like a chair does.]
>
> 2. I believe that the Goldbach conjecture (that every even natural number
> > 2 is the sum of two primes) is a meaningful statement that has a definite
> truth value (even if we may never discover this truth value and even if the
> Goldbach conjecture turns out to be independent of ZFC).
>
The word "exists" has a definite meaning only with respect to the the
theory T in which it is used. Thus if you say that the natural numbers
exist as an axiom in Peano Arithmetic (PA), it only means that it is
legitimate to talk about natural numbers in PA, and you are allowed to
construct propositions reagarding natural numbers in PA. But what if,
say, Goldbach's conjecture (GC) is neither provable nor refutable in
PA? If you are one of those who believe that a truth value for GC
"really" exists, independent of any axiomatic theories constructed by
the human mind, then you are a hard-core Platonist (Goedel himself was
one); in this case, the "existence" of such a "reality" can only refer
to some Platonic world, in which natural numbers "really exist" and GC
is "really" either true or false, independent of the human mind.
In my honest, humble opinion, this is a completely unacceptable
philosophy, no better than science fiction. If GC is genuinely
undecidable in PA, then its truth value can only exist as axiomatic
constructions in the human mind in the theories PA+GC/PA+~GC which are
*interpretations* of the theory PA, as my proposed logic NAFL asserts.
As my preprint PITT-PHIL-SCI00000635 shows, this leads us to the
conclusion that if the NAFL version of PA is consistent, then GC
cannot be undecidable in PA. Thus any (metamathematical) proof of the
undecidability of GC in PA is also a proof of the inconsistency of PA
in NAFL.
Sincerely,
R. Srinivasan srad...@in.ibm.com
Aatu Koskensilta
Bill Taylor wrote:
> Aatu Koskensilta <aatu.kos...@xortec.fi> writes:
>
> |> By silly formalism I refer to the strange caricature of formalism
> |> usually presented in the prefaces or introductory chapters of books on
> |> foundations of mathematics; that mathematics is merely a game of symbols
> |> and so forth.
>
> This is indeed silly, but I wonder whether in fact, has any mathematician
> actually *espoused* this view? I tend to doubt it.
I know of none.
> It strikes me as being
> an (as usual) only partly understood and furtherly dumbed-down version by
> various popularisers. And I think vital parts have been left out, though
> I can't say exactly what, not being all that au fait with "official" formalism.
It seems that usually everything that actually makes formalism a
sensible position is left out. When reading these texts, I always
wondered why has or would anyone actually believe such a silly thing. Of
course, later reading of texts by Hilbert and co as well as later
formalists revealed that no one actually has, nor in any likelihood
will, believe such silly things.
> |> The original formalism of Hilbert and co is bit more subtle and
> |> sensible; certain propositions are held to be meaningful ...
> |> ... whilst other are held to be idealistic -
> |> - tools for producing results about these finitistic objects.
>
> This actually strikes me as being a remarkably sound and defensible view.
Which it is.
> AFAICUI it seems to say: yes the naturals are *really* there, in ontological
> space, and some things are definitely true about them, and that PA is among these.
This *could* be a part of a formalistic philosophy of mathematics, but
is not necessary: you can hold "finitistic" statements about, say the
natural numbers, to be meaningful without espousing mathematical realism
or platonism.
> And more precisely, it gives a very hard-edged criterion for the "meaningfulness"
> of arithmetical (number-theoretic) propositions. In that when P is a decidable
> (i.e. recursive) property of naturals, & "n" a numeral, then statements of form
>
> P("n") are fully meaningful - can be tested either way;
>
> Ax P(x) are falsifiable-if-false, (but not verifiable), I dub them "scientific";
>
> Ex P(x) are verifiable-if-true, (but not falsifiable), I dub these "religious";
>
> AE or EA are neither falsifiable nor verifiable; and thus actually meaningless
> in themselves, but acquire meaning when used in logical conjunction
> with the earlier sort, especially the first type.
Yes, and you're actually building the arithmetic hierarchy here (or
something very closely resembling the arithmetic hierarchy anyway).
It should be added that it is a practical impossibility to do
mathematics with only "finitistic" elements without resort to the
"ideal" tools and constructs; your proofs simply become too long
sometimes (and not linearly, in fact the speed in which this can happen
exceeds that of any recursive function).
> It's not unlike the situation in science whereby there are many anti-positivistic
> concepts still held to be meaningful because of their untanglable relation to
> mere "observation statements". So in science the positivists were silly;
To an extent, yes. Partly they too have fallen prey to the trend of
portraying "lost causes" in philosophy in somewhat unflattering and not
necessarily all that accurate ways. Otto Neurath's anti-foundationalist
stance, for example, has always striken me as very sensible.
> the more inclusive view of e.g. Deutsch, which includes "explanations" as
> almost coeval with observations, is more sensible. Similarly a strictly
> finitist view of math, that only decidable propositions are meaningful,
> (?"strict finitism"?), is pretty silly. It should at least extend to
> statements of the 2nd & 3rd types, which are semi-decidable, and ultimately
> (a la Hilbert) to the 4th, by considering "internal consistency" and their
> "untanglability" from the rest.
Yes. Just as science would be impossible if all it was to produce were
statements about particular macro objects and their behaviour.
> So basically I'm with Hilbert on this.
>
> And I don't, and I don't imagine that Hilbert did, extend this to ZFC with
> full confidence - certainly not to ZFC + LCs. And I'm not all *that* sure
> about impredicative analysis, come to that.
Hilbert viewed set theory and the Cantorian paradise (which he *did*
have trust in) as highly useful ideal tools. The difference to the
picture you're presenting is that ZFC is not taken to be a theory of
sets, but as a tool of mathematics for producing interesting results
with "verifiable" consequences.
> So I think we could be called "limited realists" - we're committed to the
> realism of N (and its equivalent minor extensions), but increasingly more
> agnostic the further away from it one gets.
I still insist that the stance that "finitistic" statements are
meaningful does not force one to adopt a realistic (Platonistic)
semantics for mathematics.
John
Granted that a subject-predicate assertion is *about* its logical
subject, if there are no non-existents, there is nothing for "God
exists" to be about (that is, if there is no God). Nevertheless,
whether it is true or false, "God exists" is--as Peirce would probably
grant--*about* God. By allowing that there are both identical-with-
somethings and identical-with-nothings--and by equating identity-
with-something with existence (in a relevant domain, which could be
fictional: Sherlock Holmes exists in fiction, but not outside
of fiction), and identity-with-nothing with failure to exist
(in a relevant domain), one makes sense of Peirce's intuition
that there is *always* something for subject-predicate
propositions to be 'about': that which is signified by their
logical subject, which may or may not be an identical-with-
something. Indeed, logicians who have wanted an existence
predicate, have generally availed themselves of (i) or (ii)
as definitions.
(i) "exists x" means "Ey(x=y)"
(ii) "exists a" means "Ey(a=y)"
So, the equation of existence with identity-with-something is
nothing new. What is new is to allow into the domain (Peircean?)
entities which satisfy ~(exists x)/~Ey(x=y), and in so doing,
eliminate the presumption that (Ex) has 'existential force'.
--John
PS Charles Crittenden's "Ontology and the Theory of Descriptions"
(_Philosophy and Phenomenological Research_, Volume 31, Issue 1
(Sept. 1970), 85-96) mounts compelling arguments against the
presumption that (Ex) has existential force.
Aatu Koskensilta
Bill Taylor wrote:
> I write here to largely agree with Aatu Koskensilta <aatu.kos...@xortec.fi>
> who writes:
>
> |> There have been some attempts to derive the axioms of ZFC from certain
> |> natural axioms pertaining the cumulative hierarchy.
>
> True. I find the Dana Scott approach most congenial. A much underrated paper.
>
> |> However, I don't
> |> think you can ever have a "definition of set" in the sense you seem to use.
>
> True. In effect, sets are defined by their use in math, or more basically
> by their axiomatics in ZF(C), thought there are people (hello Torkel!) who
> object very vociferously to this latter viewpoint.
I agree with you in that the informal notion of set is defined by its
use in ordinary mathematics. I don't see what saying that it is
"defined" by the ZFC axiomatisation adds to this. Similarly, I don't
consider the concept of natural number being "defined" by PA. Of course,
the system of natural numbers *is* precisely what is defined by the
second order PA, but that's a whole another matter.
> |> Talk about sets does not commit one to platonism
>
> Perhaps; but as Mike Oliver often notes, working with them *does* so commit one,
> at least temporarily. And it's hard for most of us to work with sets but
> simultaneously merely "pretend" that they exist.
I don't agree with this. My opinion is that Platonism *is* necessary if
one insists on having model theoretic semantics for ordinary
mathematical language, but that this semantics is not likely to be the
"real" semantics for natural language (or even the mathematical fragment
thereof).
> |> > (1) We don't know if ZFC has _any_ models at all.
> |> > It might be an inconsistent theory.
>
> |> Well, in simlar vain it is possible that PA is inconsistent.
>
> Sternly resisting the temptation to do a spelling flame, (and indeed it is
> such a serendipitous one it may well be intended!), I nonetheless note that
> this is not really a fair comparison.
Ok, it's not a fair comparison. :)
> To conceive of the possibility of PA being inconsistent is fair game for
> ridicule, IMHO. But to conceive of an inconsistency in ZF is not nearly
> so heinous; after all the alleged model is *nowhere* near as viewable and
> verifiable as is N for PA.
If we have an intuitive notion of the cumulative hierarchy, then it
seems rather trivial to notice that the comprehension and replacement
axioms are satisfied in it (the rest of ZF are quite trivial). This is
not all that much more complicated than noting that N models PA. But
you're right, it is not as evident as the fact that N models PA.
> I myself do not for a moment expect ZF ever to
> turn out to be inconsistent, but it is not a laughable idea. We do not
> really grok V all that well! (OK so I speak for myself, but I can maintain
> a reasonable suspicion of those who blithely claim otherwise.)
I'm in agreement with this. However, I do have almost as much confidence
to the cumulative hierarchy, or at least the portion of it required for
ZFC to be consistent (i.e. up to the first strongly inaccessible cardinal).
> |> We have it in a sense backwards these days. Arguable for natural
> |> numbers, and to a much lesser extent for sets, the original intention of
> |> the axiomatisation was to characterise precisely (up to isomorphism)
> |> what the structure we study is and what properties it haves by virtue of
> |> being just that structure.
>
> I do agree with all that, however.
> (I might interject "alleged" structure for V, but only when being bloody-minded.)
>
>
> |> > It seems illogical to interpret the theorems of set theory
> |> > (ZFC) in a set (equipped with a binary relation, which is itself a set)
> |> > itself. Maybe not illogical, but certainly strange.
>
> This earlier complaint is often made, and is a fair one. I think it is
> simply answered by noting that the amount of set theory required for
> a meta-theory of the models for a logic is FAAAAAAaaaaar far less than
> the amount given to us by ZF and its V. The set-iness you have to swallow
> for merely modelling FOL is quite digestible.
True.
> |> I find the view that logic is somehow the "foundation" of mathematics
> |> rather odd; one needs quite an elaborate mathematical machinery to even
> |> get to the interesting results of mathematical logic.
>
> Exactly so; as in my parallel post. N founds logic, not the other way round!
--
Mitchell Smith
Leonard Blackburn wrote:
> whop...@alpha2.csd.uwm.edu (Mark) wrote in message news:<an017q$st6$1...@uwm.edu>...
> > blac...@math.umn.edu (Leonard Blackburn) writes:
> > >When I was younger, I was told that a set was a "collection of
> > >objects".
> >
> > There's nothing more complicated to it than that.
>
> There must be! Otherwise, the collection of all sets that
> do not contain themselves would be a set.
>
Get out your Jech and look at the axioms of ZFC. To a large extent, they are
characterizing closure properties on the class universe. They are not, in my humble
opinion, expressing the intuitions which comprise my understanding of a collection.
Now, ask yourself what the class universe is, what the class universe might be, and
how the elements of the class universe might possibly have the same object type as the
class universe.
The class universe is, most definitely, a partial order. Even the possibility of
proper classes different from the class universe are captured in this
characterization.
The axiom of union is "almost" an axiom of arbitrary union. The axiom of pairing and
the axiom schema of separation "almost" provide for finite intersections. So,
perhaps, we can think of the class universe as "almost" being a topology.
Let's go for it! Since Russell's paradox precludes separations of the class universe
into complements using the axiom schema of comprehension, let us take the class
universe as a connected topology for which every element is open as a subclass.
Next, observe that the axiom schema of separation effectively precludes a language
user from proving that any proper subspace which is a set is not totally disconnected.
So, instead of thinking about collections, try thinking about totally disconnected
topologies. And, when considering the large/small distinction associated with proper
classes, think in terms of connectedness and disconnectedness.
There is one other really nice feature in this modification. When you draw a Venn
diagram expressing "x is an element of X" you probably draw a simple closed curve
which separates the plane of your paper into an interior region and an exterior
region. That is, your abstract representation of a set has a precise topological
characterization that reflects a mutually exclusive logic.
How convenient.
:-)
mitch
Thad Coons
news:aa503d8.02092...@posting.google.com...
> When I was younger, I was told that a set was a "collection of objects"
> definition). I later read similar definitions by famous mathematicians
> that refined this definition, giving what we generally call the
> axiom schema of full comprehension, and I learned that these definitions
> were invalidated by the discovery of Russell's Paradox (and other
> paradoxes). Then I learned about axiomatic set theory. First, I was
> excited because I thought: "Finally, I will get an airtight definition
> of the word _set_." But after reading a little book on set theory by
> Charles Pinter, the first chapter of _Set Theory_ by Thomas Jech, and
> most recently, the first two chapters of _Set Theory_ by Frank Drake
> (as well as a paper on the Continuum Hypothesis by Carol Karp and a
> couple of other papers), it slowly dawned on me that I still don't know
> what a set is! The axioms of ZFC do not provide me with a definition,
> they merely attempt to say true things about some objects called _sets_
> which are left undefined. Drake says:
I'm afraid I'm late to this party. I looked into these studies on my own
a few years ago, trying to understand what was said without going too deeply
into the technical details.
The idea of a set being a collection of objects is intuitively fairly
obvious, but it requires some clarification in order to be mathematically
useful. This is set theory's equivalent to Euclid's "definition" of a point,
"That which has no part". It may be useful as a description, but it doesn't
work as a definition.
What is an object? What is a collection? We may assume that set is a
collection of objects that is to be treated as a single object; otherwise
there is little point in even considering them. What properties may sets
have? How are sets related to objects in general? To themselves? To each
other? What operations on objects yield sets, what operations on sets yield
objects, and what operations on sets yield sets? Can these relations and
operations on sets be defined in terms of relations betweem objects and
between each other? The various axiomatic theories do not consider all the
plausible answers to these questions or compare the alternatives.
It would be useful to have a single well-defined conceptual structure
that can address all of these questions and is clearly and consistently
related to other areas such as logic, arithmetic, algebra, and geometry,
but I've never seen anyone put it together. People working in mathematical
foundations tend to be work on the upper stories of sometimes very technical
theories instead of reexamining the foundations of those theories.
Thad Coons
Mitchell Smith
Leonard Blackburn wrote:
> Mitchell Smith <mit...@enteract.com> wrote in message
>
> > What follows is the first exposition of my personal research in class theory.
>
> I look forward to reading it when I have more time.
Feel free to contact me with questions.
:-)
mitch
Charlie-Boo
> Does anyone know where I can read more about exactly what sets are? Have
> there been attempts to refine the description of the cumulative type
> structure of sets since Drake's book.
> Also, every professor I've ever had in college and now in graduate school
> has used the word "set" frequently in his or her lectures. Does this mean
> that they are all Platonists? In fact, I have never met a student or
> professor of mathematics who doesn't talk about sets. Were they all Platonists?
> Or is talk about sets sometimes just abbreviated talk about well-formed
> formulas of set theory?
> Furthermore, in every set theory book I have looked at (the ones
> listed above as well as a book by Kunen, and a couple of others), there
> is always some talk about models of ZFC, where by model one means
> a structure that has a set as an underlying universe and a binary relation
> on that set that interprets the membership relation symbol.
> Thanks in advance for any helpful comments.
>
> Leonard Blackburn
> Student, University of Minnesota
Leonard,
First, I would throw away the books and articles written by your
pseudointellectual professors - especially any that talk about ZF or
ZFC. ZFC is for people who don't know how to resolve the Liar
Paradox, and so they just make up assumptions and call them "axioms".
One of the most ridiculous results is that they confuse axioms with
true statements and "prove" theorems based on axioms that themselves
are never proven.
Now, people may say, "You have to have undefined axioms.", but that is
not true. For at least 2,000 years people have bought Euclid's
argument and the idea that the axioms shouldn't be provable or
actually true. But in reality, they are always based on something
that is true that is being modeled. This is especially true if you
then start "proving" that certain statements are true (e.g. in ZFC.)
If any of those eggheads in their ivory towers ever faced the real
world, they would become a lot smarter from seeing the problems,
solutions and ideas that are developed from real life problems.
But back to the original question: What is a set? Well, the fact is,
all of Mathematics is constructed from the natural numbers N = {0, 1,
2, ...} and {true, false}. These are the basic finite and infinite
concepts. And from them we can define the rationals, real numbers,
imaginary numbers, transcendental numbers, algebraic numbers, etc.
But in each case, we are just considering some combination of the
primitives. And when the combination is "map N to {true, false}" then
we call it a "set".
Now, the concept of a set has real life analogues. But that is to be
expected, as all of Mathematics is modeling something. In fact, it
can be argued that any comparison to the real world must show a
correspondance, because we are merely comparing Mathematics in general
to Mathematics in particular.
Charlie Volkstorf
Cambridge, MA
www.arxiv.org/html/cs.lo/0003071
PS I developed my ideas from analyzing real world problems as a
computer programmer for years. As an example, one of my first
projects was to develop a system for generating reports automatically.
I developed a formalism that I later discovered was the 1st order
Predicate Calculus, except that my syntax is not ambiguous. You see,
Computer Science is just a generalization of Mathematics, and by
having to actually formalize and, more importantly, have those
formalizations verified by programming them, we build much more
general and precise systems than the Math/Logic/Philosophy professors
ever dreamed of.
Mitchell Smith
Thad Coons wrote:
> What is an object?
Admittedly, this is an odd suggestion, but may I recommend that you look to the
philosophy of Immanuel Kant for this answer (He does, by the way have opinions
about the foundation of mathematics which are not entirely reflected in modern
foundations).
This is from a recent correspondence in which I was engaged:
------------------------------------------
The process of concept formation described in the "Critique of Pure Reason" is
essentially a checkpointing process:
"For this unitary consciousness is what combines the
manifold, successively intuited, and thereupon also
reproduced into one representation."
"...the unity which the object makes necessary can be
nothing else than the formal unity of consciousness
in the synthesis of the manifold of representations.
It is only when we have thus produced synthetic unity
in the manifold of intuition that we are in a position
to say that we know the object."
Compare this with the ACID properties which govern transaction processing in
computer operations:
Atomicity - A transaction's changes to the state are
atomic: either all happen or none happen.
Consistency - A transaction is a correct transformation
of state. The actions, taken as a group
do not violate any of the integrity
constraints associated with the state.
Isolation - Even though transactions execute
concurrently, it appears to any given
transaction that other transactions
executed either before or after the given
transaction, but not both.
Durability - Once a transaction completes successfully,
its changes to the state survive failures.
Durability is reflected in "the unity which the object makes necessary."
Consistency is reflected in "the formal unity of consciousness." Isolation is
reflected in "successively intuited." And, atomicity is reflected in the last
statement of the second quote: "It is only when we have thus produced synthetic
unity in the manifold of intuition that we are in a position to say that we know
the object."
-------------------------------------------------------------------
> What is a collection?
In my humble opinion, a collection is precisely characterized by a totally
disconnected topology. Actually, the concept I am thinking of is described as
totally separated in Steen and Seebach, but the idea is the same. The
granularity of topologies range from that of a partial order whose only atom is
a big lump to a partial order whose atoms correspond to singletons. The
intuitions underlying collections arise from the latter.
> We may assume that set is a
> collection of objects that is to be treated as a single object; otherwise
> there is little point in even considering them.
There is an ambiguity here. Clearly, the utility of sets arises from the fact
that a collection is ascribed object semantics. But, what is meant by object
semantics? When I asked myself that question, I began to think about topologies
as objects and totally disconnected (or, totally separated) topologies as
objects having collection semantics.
> What properties may sets
> have?
I believe it is improper to begin with object-based intuitions. In my personal
research, I have focused on how the representations for sets are manipulated in
order to convey my intuitions concerning sets. This has led to consideration of
the "circle" representations and the "dot" representations used in Venn
diagrams. "Circle" representations reflect collection semantics. "Dot"
representations reflect object semantics.
How is this useful?
Well, in a general first-order algebraic structure the only admissible "circle"
representation is that corresponding to the universe of discourse. Moreover,
this representation reflects the role of the membership predicate in its
metalinguistic usage.
It it true that we like to think of a correlation between subclasses of our
universe of discourse and the collections of term referents which satisfy given
sentences of our language. This well-established correlation, however, reflects
an implicit extension of the first-order algebraic structure to a topological
structure.
A topological structure extends the language of the first-order algebraic system
to include class variables with collection semantics and the membership
predicate to relate the constituents of the model domain to the referents of
these new variables. With suitable constraints with respect to formula
formation, a topological structure can be given the semantics of a topological
model.
In any case, the point is that if we proceed directly to considering object
properties of sets, we obscure subtle usage rules concerning the predicates we
use to convey our intuitions.
> How are sets related to objects in general?
Naively, sets are collections of objects.
In Kantian philosophy there is a twist on this. Object formation is
accomplished by the understanding. The rules for concept formation are applied
to a manifold of sensible intuition which is related to a separate faculty of
knowledge from that of the understanding. So, some construct is taken to precede
the notion of object.
Multiplicity associated with this manifold is suggested by
"Whereas all intuitions, as sensible, rest on affections, concepts
rest on functions. By 'function' I mean the unity of the act of
bringing various representations under one common
representation."
So, one could almost conclude that sets precede objects.
This would not be correct, however. A later philosopher, Husserl, developed
intuitions concerning formal ontologies. The theory governing formal ontologies
is called mereology. It is based on a reflexive order relation expressing the
predicate "is a part of." Both Husserl and Cantor studied under Weierstrass and
were acquainted with one another as well as with each other's topological ideas.
The mereological predicate which can found a topology is the strict transitive
relation expressing the relation "is a proper part of." It is intended to be
coincident with "is a part of" except for situations where the latter is
satisfied reflexively. For comparison purposes, you may think of "is a proper
part of" as proper set inclusion except that you should not make the mistake of
thinking that "is a proper part of" is logically equivalent to proper set
inclusion. If you require evidence for this, look at Cohen's forcing language
and the Generic Model Theorem. These results are possible only because of the
ambiguity associated with these order relations.
Note, however, that mereotopology has somewhat different characteristics from
topological intuitions expressed through sets. The unary predicate "is a point"
is defined by
forall x ( Pt(x) iff (forall y (y part x implies y equiv x) ) )
You may compare this with the multiplicity expressed by Kant when describing the
functionality associated with concept formation.
> To themselves?
This response is already getting too long. However, it is to be noted that a
first-order algebraic structure whose term referents are hereditarily defined
classes expresses a topological structure without the need for extending the
language.
This is why it is important to avoid considering object-based properties first.
Well-established usage is not necessarily rigorous.
> To each
> other?
The axiom schema of separation suggests that the constituents of the class
universe totally separate one another.
> What operations on objects yield sets, what operations on sets yield
> objects, and what operations on sets yield sets?
These questions are not necessarily well posed because of the differences
between collections whose constituent objects present collection semantics,
colllections whose objects do not present collection semantics, and collections
whose objects present a mixed profile with respect to collection semantics.
> Can these relations and
> operations on sets be defined in terms of relations betweem objects and
> between each other?
Can they be defined at all? Formal systems in mathematical logic rely on the
extensional interpretation of language elements. In my humble opinion, this is
completely at odds with the conclusion which must be drawn from even the
simplest use case analysis directed toward discerning how mathematicians use
language.
Mathematicians define their terms.
But, before we can ask about implementation through definition, we must know
whether or not there is any reasonable way to use a given formal language with
intensional semantics--that is, as if it were a natural language.
> The various axiomatic theories do not consider all the
> plausible answers to these questions or compare the alternatives.
That is our job, is it not?
>
> It would be useful to have a single well-defined conceptual structure
> that can address all of these questions and is clearly and consistently
> related to other areas such as logic, arithmetic, algebra, and geometry,
> but I've never seen anyone put it together. People working in mathematical
> foundations tend to be work on the upper stories of sometimes very technical
> theories instead of reexamining the foundations of those theories.
>
"To know what questions may reasonably be asked is already
a great and necessary proof of sagacity and insight. For if a
question is absurd in itself and calls for an answer where none
is required, it not only brings shame on the propounder of the
question, but may betray an incautious listener into absurd
answers, thus presenting, as the ancients said, the ludicrous
spectacle of one man milking a he-goat and the other holding
a sieve underneath."
--Immanuel Kant
I like your questions.
:-)
mitch
>
> Thad Coons
Bill Taylor
|> As an example, Gerog Kreisel has found a proposition p, of which the
|> shortest proof in FOL has more steps than the visible universe has
|> particles, but which has 5 step proof in second order logic.
Aatu - can you expand on this a bit please? I've dimly heard of such
a thing, but forgotten what it was. Can you please outline Kreisel's
(or any other) example for us?
--------
Regarding the other thread, we seem to agree pretty much on the scope of
a proper view of formalism/finitism; but there's just one point of disagreement
which didn't seem important enough for a separate reply.
You say that a commitment to the meaningfulness of decidable statements
does NOT entail a commitment to a realist view of natural numbers. As you note,
I disgree, but I suspect we're disagreeing over mere philosophical definitions,
and not anything serious. So I doubt there's any need to continue on this one
- it is not a math-philosophical disagreement, merely a linguistic one, I suspect.
Cheers! Nice to see you joining the group!
------------------------------------------------------------------------------
Bill Taylor W.Ta...@math.canterbury.ac.nz
------------------------------------------------------------------------------
"Dr Strangelove" includes a Russellian auto-destruct device.
It blew itself up!
------------------------------------------------------------------------------
Bill Taylor
|> Do you mean Boromir the character from _The Lord of the Rings_?
The same. From the film, not the book. He says derisively at one point:
"Gondor has no king; Gondor *needs* no king!"
I like to transpose it to math-logic: PA has no foundation, PA *needs* no...
|> But the principle of natural induction is
|> really about *collections* of numbers, rather than numbers. So I
|> think that there is a tiny bit of set theory present in even the
|> most elementary number theory.
You say tiny, so I can hardly accuse you of going overboard. But really,
I think it so tiny as to be invisible. We can couch induction without any
reference to sets, as I have in an occasional sig:-
"Successorial properties hold universally if initially!"
Any reference to sets is so oblique as to be almost invisible. The idea of
a property does not *necessarily* entail a set, I fancy, as is the case here.
"Universally" could be deemed to refer to the "set" of all naturals, but no
other set or subset is adduced or alluded, so I don't think it's in any way
indispensable. No more so than e.g. declaring that associativity of addition
is universal.
So, to nit-pick horribly, (or maybe counter-nitpick), I think I disagree with you.
------------------------------------------------------------------------------
Bill Taylor W.Ta...@math.canterbury.ac.nz
------------------------------------------------------------------------------
And God said
Let there be numbers
And there *were* numbers.
Odd and even created he them,
He said to them be fruitful and multiply,
And he commanded them to keep the laws of induction.
------------------------------------------------------------------------------
Charlie-Boo
I am afraid that all of the mumbo-jumbo below misses the whole point
(besides being just silly.) The concept of a set is very primitive
and universal. It doesn't take 3 pages to explain it! It is a
manifestation of a simple, general concept or two. And think about
it. The idea of a set - is this a complex matter? Of course not. It
is not complexity that makes it difficult for many people to define.
It is actually the opposite. The concept of a set is so simple and
primitive, that people have a hard time just formalizing what it means
intuitively.
(If a set were really as complex as you make it sound, it wouldn't be
used throughout Mathematics. It would only be used in those
infrequent cases where all of its various properties/requirements
occurred.)
The fact is, everything in Mathematics begins with the natural numbers
and {true,false}. The rest is just taking various combinations of
these primitive concepts. A set is just a mapping to {true,false},
either N=>{true,false} (N being the primitive recursively emumerable -
i.e. representable in the system - concept) or some combination of N
and {true,false} mapped to {true,false}.
How you interpret it is up to you: as a set, a predicate, or even as a
single real number (each bit (instead of digit) being defined by
whether or not a given value is in the set.) But all it really
represents is a mapping (primitive or otherwise) to {true,false}.
Beyond that you are just further manipulating these parts that make up
the definition.
Got it? Good.
Charlie Volkstorf
Cambridge, MA
www.arxiv.org/html/cs.lo/0003071
> The process of concept formation described in the "Critique of Pure Reason" is
> essentially a checkpointing process:
>
> "For this unitary consciousness is what combines the
> manifold, successively intuited, and thereupon also
> reproduced into one representation."
What a bunch of mumbo-jumbo! "unitary consciousness"? "successively
intuited"? Are these even real words? What a joke!
> "...the unity which the object makes necessary can be
> nothing else than the formal unity of consciousness
"The formal unity of consciousness"? LOL! Do you really buy this
nonsense?
> in the synthesis of the manifold of representations.
> It is only when we have thus produced synthetic unity
> in the manifold of intuition that we are in a position
> to say that we know the object."
>
> Compare this with the ACID properties which govern transaction processing in
> computer operations:
That's it - the guy who wrote that must have been on ACID.
> "It is only when we have thus produced synthetic
> unity in the manifold of intuition that we are in a position to say that we know
> the object."
Ok children, how many sentences can we make up using the words
"unity",
"manifold" and "intuition"?
> > What is a collection?
>
> In my humble opinion, a collection is precisely characterized by a totally
> disconnected topology.
Humble opinion? Ha! You're not being humble. You're trying to look
like a big pseudointellectual!
> In Kantian philosophy there is a twist on this. Object formation is
> accomplished by the understanding. The rules for concept formation are applied
> to a manifold of sensible intuition which is related to a separate faculty of
> knowledge from that of the understanding. So, some construct is taken to precede
> the notion of object.
Kantian philosophy? What a name-dropper! Who do you think you're
impressing with this nonsense? What did this ever accomplish? (And
please, no more mumbo-jumbo!)
Charlie Volkstorf
Cambridge, MA
www.arxiv.org/html/cs.lo/0003071
> I like your questions.
>
> :-)
>
> mitch
> > Thad Coons
Bill Taylor
|> There's not too much point in rehashing this as we've been over it so many
Oh of course; d'accord!
I was not seeking to have any "last words"; so much as re-stating, for the benefit
of the many newbies who seem to have joined us recently; many of them fine-sounding
fellows! It must be the academic new year in the north?
|> I do not see any sharp lines between the
|> completely abstract and the completely concrete;
Indeed. But I'm not 100% clear on your "concrete". So I would like to hear you
answer:- Do you see a sharp line beteen the abstract and the *physical* ?
|> Clearly there *would* be a rational
|> response if arbitrarily large naturals were refuted,
Again, I cannot discern what this means, but I pose no counter-question, so
please feel free to drop it. I merely restate: I find it inconceivable that
arbitrarily large naturals could "be refuted". I put it on a par with having
god and his angels appear cutting through the sky on a fiery chariot. Less so.
---
|> I keep looking for a good opening to expound my thoughts in detail
|> on monotonic versus nonmonotonic knowledge;
Mike - you keep teasing us with tidbits about this! I wish you would come clean.
Now OC I don't want to rush you into anything incompletely cooked, but may I
at least reasonably request an advance preview? A trailer for the next film?
Maybe you could give us just a short paragraph only, outlining in horrible
brevity, just what sort of thing non/monotonic knowledge is ABOUT? So at
least we're a little more prepared, and might mull over some thoughts ourselves
in advance.
Priority will still be safe with you! ;-)
|> Actually my epiphany vis-a-vis informal ordinals came from reading
|> the "Birthday Cantatatata" in _Goedel_Escher_Bach_,
A wonderful book. Not perfect, but certainly a grand dipper-into.
I rechecked "Birthday Cantatatata" and found it very good too.
|> *long* after I was already quite familiar with their properties.
I already knew of the matter, so had no epiphany thereby. And I learned of
ordinals constructively rather than "algebraically", i.e. via their universal
properties as you did. Or at least, alongside. But I too count it as one
of my "grand eye-openings" into the world of ontological space (q.v.) :)
Not my very greatest, but certainly one of the top 3 or 4.
So Hi!!
---------------------------------------------------------------------------------
Bill Taylor W.Ta...@math.canterbury.ac.nz
---------------------------------------------------------------------------------
The nature of light is "?" .
The upper part denotes the wave aspect, the lower part the particle aspect.
---------------------------------------------------------------------------------
Thad Coons
news:3D9B9FBD...@enteract.com...
<snipped>
I must confess that most of what you wrote was far over my head. What I
know about Kant might amount to a page, what I know about topology is
scarcely more, I couldn't tell a first-order algebraic structure from any
other order to save my life, and I have only the foggiest notion of what a
"membership predicate in its metalingistic usage" might mean. I'm not
computer scientist enough to know anything about transaction processing, or
logician enough to look at either Cohen's forcing language or the Generic
Model Theorem.
Where you said "disconnected topological structure", I thought of
Discrete and separate objects", for instance.
I'm asking questions at an earlier and less formalized stage of thinking
than you seem to be using. For myself, I would examine how sets are used and
try to identify inconsistencies and ambiguities in natural language by
appropriate definitions before I tried defining properties or relationships
with more formalized logical predicates.
The reason I mentioned properties of sets is that set theory has
advanced far enough that we can distinguish several different kinds of sets.
For instance, there are theories of finite sets, infinite sets, unordered
sets, ordered sets, sets with duplication, fuzzy sets (not well-defined),
self-inclusive sets, and sets with various kinds of topological structure
(discrete, continuous, and so forth). Each of these general classes or
categories of has properties like, but unlike the others.
I mentioned relationships because there is the membership relation, and
its naive use which led to the shock of Russel's paradox, and the subset
relation. I have encountered mereology, and I'm aware that "is a part of" is
not the same as either "is an element of" or "Is a subset of". My hand is a
part of my arm, but I'm not accustomed to thinking of it as either an
element or a subset of it. The relationship of mereology to set theory
certainly ought to be examined in mathematical foundations.
The relationship of logical predictates to the objects or collections
they are being used to describe is another question. The normal usage seems
to imply a connection with set theory, but the possibilities of different
kinds of sets combined with the different varietes of logic suggest that the
relationship ought to be re-examined occasionally.
Thad Coons
Bill Taylor
|> {1,2} = {2,1}, (comma is commutative)... (comma is idempotent).
|> comma is associative.
|> Evidently the comma can be seen as a mathematical operator with
|> algebraic properties that aggregates elements into a structure that
|> is simpler, more primitive, than sets;
What a wonderful idea! I've noticed for a while now, e.g. in CS "lists",
that the lowly comma was being used as an operator.
|> curly braces can be seen as an operator that applies to a bunch and makes a set;
...
|> Bunch theory tells us about aggregation;
|> set theory tells us about packaging. The two are independent.
So we are breaking down the formerly thought-to-be ineluctably primitive
"set" idea into two sub-ideas. How marvellous. I certainly hope something
sensible can be made of it. And with more excitement but less optimism,
that some new mathematical fruits result from it.
I shall certainly be looking at the Hehner article. Not just a Playboy, clearly!
I like the idea of packages - it certainly would make the empty set more
accessible to the undergrad-in-the-street; a package with nothing inside it,
like you sometimes get at not-so-serious birthday parties.
And as the student-in-the-street often feels that there is no difference
between an object and its singleton, maybe this idea could be given some
respectability too. e.g. that {1,2} n {2,3} = 2, rather than {2}.
As the man-in-the-street (or even Quine!) might insist.
Well anyway, perhaps nothing much will come of it all, but here's hoping!
------------------------------------------------------------------------------
Bill Taylor W.Ta...@math.canterbury.ac.nz
------------------------------------------------------------------------------
Basic set theory is a shotgun marriage between aggregation and packaging.
The two parties get along OK - but they hardly seem made for each other.
------------------------------------------------------------------------------
Bill Taylor
|> 1. I believe that N (the set of natural numbers) "exists" as a completely
|> coherent idea and that <N,0,s> is a model of the Peano axioms.
IMHO this is the essence of a "realist" view of math. (Make Platonism a subset
of realism if you like - I don't really think it exists any more.)
|> 2. I believe that the Goldbach conjecture (that every even natural number
|> > 2 is the sum of two primes) is a meaningful statement that has a definite
|> truth value
Likewise. But note that if it is independent of ZFC, it is thereby automatically
*true* in N; by virtue of its particular form - universal quantifier only.
|> 3. I believe that there exists a coherent notion of "set," that the
|> cumulative type hierarchy of sets (V) is a coherent idea that mathematicians
|> have begun to describe (even if the description isn't complete), and that
|> the axioms of ZFC are true statements about V.
Most pure mathematicians would include this as a prerequisite for realism.
I don't go so far; I feel N/PA is sufficient.
|> Feel free to add any others
|> that may help in clarifying the definition of "platonist" for me.
Most mathematicians would include analysis. This was most decidely NOT
part of realism before Newton/Euler. "The great Berk", bishop Berkeley was
perhaps the last dying kick of the horse who refused to accept its reality,
and since Weirstrass it has been virtually unassailable, except perhaps
for lingering concerns over effectiveness and predicativity.
|> I have spoken with other mathematicians (students) who believe that
|> the word "platonist" has a negative connotation,
It does seem to. I think this is because of its philosophical odour;
metaphysics is not popular among mathies, and the name of Plato smells a little.
That's why I prefer to call it "realism", a word that would find much more
favour with mathies and others, with its macho, in-your-face sound.
|> and that it is ridiculous to believe that mathematical objects
|> somehow exist as ideal things in some Platonic realm,
I prefer to regard them as *real* things in an *abstract* realm.
This distinguishes it from both physics (physical realm)
and the humanities (subjective things).
|> mathematical objects exist even before humans have "discovered" them.
Not just before *that*, but before the big bang, even!
|> They have likened this to faith or religion.
Not a totally unreasonable analogy, but one that does not stand up to close
scrutiny. There is never a "right answer" in theology, not even to the
simplest questions; nor is there a mechanic for finding them if there were.
More significantly, no mathie ever burnt another mathie at the stake for
believing the wrong sort of math; nor do they crash planes into big buildings
or kill masses of people in other ways such as setting fire to crowded trains!
Nor do they try to make their children feel guilty for not being mathematical
enough. Hmmmm... no maybe we should scrub that last one...
|> I was one of these people who also went so far as to think that
|> "mathematical truth" is not a coherent concept and has no place in math
Well *weren't* you a silly boy then, Leonard?! :)
|> I probably do not mean the same thing that Plato himself did.
It's doubtful that *Plato* meant the same thing that Plato did.
It's better to drop "Platonism" and stick to "realism".
|> some "location" where such objects might exist.
Of course not. Abstract objects do not HAVE a time or place. Physical
objects have those. Abstract objects DO have structure, pattern and
relationships though, and remarkably precise and clear ones, too!
If you want to "place" them, for the purposes of ease of discussion,
I recommend "ontological space", mainly because its my own baby.
|> For example, did the constellations (Ursa Major, etc.) exist
|> before humans named them?
No. Galaxies probably did though, and stars certainly did.
|> Also, I'm not sure of the difference between realism and platonism,
There isn't any, really. Platonism is a fuzzy concept, not unlike many
of its proponents. Realism is a much better term for the precision of math.
In summary, just remember that math objects are REAL, but ABSTRACT.
They have properties that are definitely TRUE regardless of people or places;
contrary to a lot of the hideous culturally-relativist deconstructionist slop
that humanities departments often serve up these days.
They live in a Popperian "4th world" - not physical, but not the 3rd world
of cultural artifacts, either. Here's a bi-dichotomous analysis...
abstract physical
.--------------------------.
| | | numbers | rocks
objective | math | science | lines | electrons
| | | |
|------------+-------------| ------------+------------
| | | |
subjective | culture | mentality | music | odours
| | | laws | pain
`--------------------------' |
------------------------------------------------------------------------------
Bill Taylor W.Ta...@math.canterbury.ac.nz
------------------------------------------------------------------------------
With or without religion you will have good men doing good things
and evil men doing evil things; but to get good men doing
evil things - for that you need religion.
------------------------------------------------------------------------------
Bill Taylor
|> Whether or not a particular idea is "completely coherent" is a matter of
|> logic. Ideas that are internally contradictory have a very weak claim
|> indeed to any sort of existence.
Well said! But often ignored in the everyday world!
|> entity "exists" iff it has a model. This of course shoves the issue from
|> the "existence" of the entity to the "existence" of the model.
Exactly. But there is still some gain in the shoving, I think.
|> Plato himself would argue that a material object such as your chair only
|> "exists" in the material universe by virtue of it being a reflection of
|> "the ideal chair" existing in the universe of forms.
Did Plato go so far as positing "ideal chairs"? If so, he's even crazier
than I thought. But I suspect you're misrepresenting him.
|> In this view, the natural numbers actually have a more solid claim
|> to "existence" than chairs do!
Godel most certainly felt this way! And as I say in my occasional sig...
"The chief difference between mathematics and physics is that
in mathematics we have much more direct contact with reality."
|> And note that even if one believes that each finite natural number
|> "exists", it is a large philosphical leap to then infer the "existence" of
|> a single infinite entity which is the collection of all the natural
|> numbers at once.
I guess that's the dividing line between a "strict finitist" and the rest.
And don't forget there are several "ultrafinitists", who hold that not even
arbitrarily large finite numbers exist, such as 10^(10^(10^10)). Even they
can still be mathematicians. Though barely math-philosophers.
|> > cumulative type hierarchy of sets (V) is a coherent idea
|> A hard-core formalist could believe that just as sincerely
Oh no, I don't think so. That's definitely going too far IMHO.
|> Saying that "there exists a coherent notion of ..." is
|> essentially a matter of logic,
No I don't think so. "Having a consistent notion of..." may be a matter
of logic; but that's not quite the same thing.
"Having a coherent notion of..." involves a lot more stuff. I'm not sure
exactly what, (no point arguing definitions), but I suggest it include
imagery, inter-subjectivity, explanatory power, and so forth.
|> Lots of things that seem "ridiculous" happen to be matters of experimental
|> fact (e.g., quantum mechanics, relativity, the fact that the earth moves
These are not really "experimental" facts, but "explanatory" facts.
Experimental facts are what comes up on the dial when you do so-and-so.
What positivists would call "observation statements". The things you
mention above are at least one level higher in the scheme of science.
|> The notion that "God is a mathematician" gained currency as more and more
There is certainly a lot of evidence for that; but I think there is even more
evidence for the case that god is an amoral experimenter who likes poking
insects to see how they wriggle.
|> A little philosophy is a dangerous thing...
Maybe. But it never knocked down any towers.
------------------------------------------------------------------------------
Bill Taylor W.Ta...@math.canterbury.ac.nz
------------------------------------------------------------------------------
Faith can move mountains, but a tanked-up Boeing is quicker
------------------------------------------------------------------------------
Mike Oliver
> Mike Oliver <oli...@math.ucla.edu> writes:
>|> I do not see any sharp lines between the
>|> completely abstract and the completely concrete;
>
> Indeed. But I'm not 100% clear on your "concrete". So I would like to hear you
> answer:- Do you see a sharp line beteen the abstract and the *physical* ?
Not clearly.
>|> Clearly there *would* be a rational
>|> response if arbitrarily large naturals were refuted,
>
> Again, I cannot discern what this means, but I pose no counter-question, so
> please feel free to drop it. I merely restate: I find it inconceivable that
> arbitrarily large naturals could "be refuted". I put it on a par with having
> god and his angels appear cutting through the sky on a fiery chariot. Less so.
Well, I don't know why you would have trouble conceiving the fiery chariot
thing. In fact it seems to me that you just did so.
"You keep using that word -- I do not think it means what you think it
means" :-)
I've given before an example of what such a refutation would mean.
Suppose you found a formal derivation of 0=1 from the theory
I Delta_0 + "10^10^10^10^10 exists". I'm quite sure you can
*conceive* of that, even if you believe for abstract reasons
that it's "logically impossible". Do you agree that we would
still have rational courses of action, and indeed that we
would still be able to use (some) mathematics?
Mike Oliver
> |> I keep looking for a good opening to expound my thoughts in detail
> |> on monotonic versus nonmonotonic knowledge;
>
> Mike - you keep teasing us with tidbits about this! I wish you would come clean.
>
> Now OC I don't want to rush you into anything incompletely cooked, but may I
> at least reasonably request an advance preview? A trailer for the next film?
>
> Maybe you could give us just a short paragraph only, outlining in horrible
> brevity, just what sort of thing non/monotonic knowledge is ABOUT?
Well, nonmonotonic knowledge is ordinary, common-sense knowledge. It's
nonmonotonic because *more* information can cause me to know *fewer*
things. I hear about a bloody glove and DNA and I know who killed
two people; I hear about a cop who had opportunity and possibly desire
to frame that person and I'm no longer so sure. (In the actual case
being alluded to I'm still pretty sure, but you get the point.)
The case of Goldbach's conjecture, in which you and I have agreed
we know the answer, is similarly nonmonotonic. Examples of new
information that would make the arguments seem less compelling:
Suppose some observed behavior of the prime numbers started
to look "nonrandom" in a very unexpected way. Or suppose (actually
this is a special case of the former) even numbers started showing
up, say below 10^100, such that none of the first trillion primes
left a prime difference when subtracted from the large number, even
if the trillion-and-first did so.
Now *proofs*, it seems to me, are an attempt to make knowledge
monotonic -- once you have a proof, it remains a proof. This
is not quite true in practice of course. Proofs are not the
same thing as formal derivations, and it happens from time
to time that an apparently valid method of proof proves something
ridiculous, an error is found in the method, and proofs have
to be revised. I'm sure it's happened to you personally.
But we conceptualize proofs as being reducible to formal
derivations even if we almost never perform the reduction,
and if you have a formal derivation, all laid out in
front of you so you can actually check each step -- well,
if *that* ever becomes a non-derivation, for reasons other
than that you just made a mistake, then it's the kind
of thing I have no rational response to. (Tie-in with
my previous post.) So that's as close as we can get to
monotonic knowledge.
But careful! that doesn't mean the *conclusion* of the
proof is nonmonotonic knowledge. It *is* imaginable that
new information will cast doubt on the soundness of the
theory, and there *is* a rational response to that.
Mike Oliver
> But careful! that doesn't mean the *conclusion* of the
> proof is nonmonotonic knowledge.
Typo -- "doesn't mean the conclusion [...] is monotonic knowledge"
Daryl McCullough
>We can couch induction without any
>reference to sets, as I have in an occasional sig:-
>
> "Successorial properties hold universally if initially!"
>
>Any reference to sets is so oblique as to be almost invisible. The idea of
>a property does not *necessarily* entail a set, I fancy, as is the case here.
Property, set, the distinction doesn't mean that much to me. It is
only when you try to formalize a theory of properties that you realize
the need for the distinction.
>"Universally" could be deemed to refer to the "set" of all naturals, but no
>other set or subset is adduced or alluded, so I don't think it's in any way
>indispensable.
My point is that induction is about second-order objects (properties,
rather than numbers).
--
Daryl McCullough
Ithaca, NY
Leonard Blackburn
> Leonard,
>
> First, I would throw away the books and articles written by your
> pseudointellectual professors - especially any that talk about ZF or
> ZFC. ZFC is for people who don't know how to resolve the Liar
> Paradox, and so they just make up assumptions and call them "axioms".
> One of the most ridiculous results is that they confuse axioms with
> true statements and "prove" theorems based on axioms that themselves
> are never proven.
First, I am a student who is about 1 year away from a PhD in mathematics.
I plan to get a teaching/research job at a college or university. As a
working mathematician, I cannot throw away ZFC (and other similar set theories),
and I cannot throw away the axiomatic method. If I do, then I throw away
any chance at getting a job. Secondly, virtually all of mathematics can
be formalized in ZFC, and _that_ is the power of set theory--as a glue
and common framework. All mathematicians use the language of sets. Nearly
all logicians/set theorists use the notions of cardinal, ordinal, transfinite
induction, definition by transfinite recursion, etc. These are all concepts
developed in ZFC. I see no reason to throw them out. In order to complete
my personal research in the area of inductive definability (which has
applications to computer science and finite model theory, including the
area around the P vs. NP problem, and which has applications to many other
branches of mathematics) I need to make use of the abovementioned aspects of
ZFC, and I need also to use the concepts of inaccessible cardinals, mahlo
cardinals, measurable cardinals, admissible sets, and other concepts. These
are all concepts of ZFC.
I see no reason to abandon ZFC and the axiomatic method in general, since
both have given us a huge amount of results in all branches of mathematics.
ZFC is just a common language. Also, the axioms of ZFC are simply
reflections of the principles of reasoning mathematicians have been using
for centuries. Mathematics is a complicated organism that has had a
very complicated historical development. Denying the language and
techniques of set theory and other axiomatic methods would be to deny
almost all of mathematics.
Granted, there are doubtless other ways to provide this glue, this common
language of mathematics, and the foundations of mathematics. But there is
no need for a switch.
Also, I don't know any "psuedointellectual professors." For instance,
my advisor is a down-to-earth guy, who is very knowledgable about
the results and history of his and related fields. He does not pretend
to know absolute truths. He does not pretend to be a philosopher. He
is simply a working mathematician--working and publishing within the
rules that history has given him.
>
> Now, people may say, "You have to have undefined axioms.", but that is
> not true. For at least 2,000 years people have bought Euclid's
> argument and the idea that the axioms shouldn't be provable or
> actually true. But in reality, they are always based on something
> that is true that is being modeled. This is especially true if you
> then start "proving" that certain statements are true (e.g. in ZFC.)
Axioms are usually written down after mathematicians have already been
assuming them for years. The writers of the axioms are not the creators
of mathematics. They are only observing what mathematicians have been
doing.
>
> If any of those eggheads in their ivory towers ever faced the real
> world, they would become a lot smarter from seeing the problems,
> solutions and ideas that are developed from real life problems.
There are many types of mathematicians. Some do see the problems,
solutions and ideas that are developed from real life problems.
Also, I don't think your portrayal of some mathematicians as ignorant
and vain is backed up very well in your post.
>
> But back to the original question: What is a set? Well, the fact is,
> all of Mathematics is constructed from the natural numbers N = {0, 1,
> 2, ...} and {true, false}.
I highly doubt that. How are the ordinals constructed from these things?
How is the notion of 'proof' constructed from these things?
Also, it seems like you believe here in the truth of an axiom: namely,
the axiom of infinity (in ZFC). Hence you are being contradictory.
> These are the basic finite and infinite
> concepts. And from them we can define the rationals, real numbers,
> imaginary numbers, transcendental numbers, algebraic numbers, etc.
How about the ordinals?
> But in each case, we are just considering some combination of the
> primitives.
What does "combination" mean?
> And when the combination is "map N to {true, false}" then
> we call it a "set".
The above sentence has no meaning for me. What is
"map N to {true, false}"? It looks like a statement in the
imperative mood. How is this to be interpreted as a combination?
If you clear this up, then I will probably have a follow-up
question: are you describing _all_ of the sets here?
>
> Now, the concept of a set has real life analogues. But that is to be
> expected, as all of Mathematics is modeling something. In fact, it
> can be argued that any comparison to the real world must show a
> correspondance, because we are merely comparing Mathematics in general
> to Mathematics in particular.
I'm not exactly sure what you are saying here, but I don't think
that an infinite mathematical set has any real life analogue.
>
> Charlie Volkstorf
> Cambridge, MA
> www.arxiv.org/html/cs.lo/0003071
>
> PS I developed my ideas from analyzing real world problems as a
> computer programmer for years. As an example, one of my first
> projects was to develop a system for generating reports automatically.
> I developed a formalism that I later discovered was the 1st order
> Predicate Calculus, except that my syntax is not ambiguous. You see,
> Computer Science is just a generalization of Mathematics, and by
> having to actually formalize and, more importantly, have those
> formalizations verified by programming them, we build much more
> general and precise systems than the Math/Logic/Philosophy professors
> ever dreamed of.
I'm sure your work as a computer scientist is good, important, and
useful. But it seems like you are condemning mathematicians for not
being computer scientists. Perhaps the Math/Logic/Philosophy professors
have different goals than you do.
-Leonard Blackburn
Student, University of Minnesota
Russell Easterly
"Bill Taylor" <mat...@math.canterbury.ac.nz> wrote in message
news:angrr2$hf8$1...@cantuc.canterbury.ac.nz...
> |> They have likened this to faith or religion.
>
> Not a totally unreasonable analogy, but one that does not stand up to
close
> scrutiny. There is never a "right answer" in theology, not even to the
> simplest questions; nor is there a mechanic for finding them if there
were.
>
> More significantly, no mathie ever burnt another mathie at the stake for
> believing the wrong sort of math; nor do they crash planes into big
buildings
> or kill masses of people in other ways such as setting fire to crowded
trains!
> Nor do they try to make their children feel guilty for not being
mathematical
> enough. Hmmmm... no maybe we should scrub that last one...
Mathies executed the guy that proved the square root of two is irrational.
In another thread, I mentioned how Gauss didn't publish his version
of a non-Euclidean geometry fearing the controversy it would create.
The latest Scientific American has an article about how Wolfram's
theory of everything is being presented to a wide audience of people
while similar theories by unknowns are ignored.
The article concludes that it is the job of the "experts" to weed out
crackpots.
I'm not sure I would agree.
Despite what mathematicians say, there is a lot of things in mathematics
that you just have to accept on faith. Math has its high priests like any
other religion.
Russell
- Zeno was right. Motion is impossible.
John
> Mitchell Smith <mit...@enteract.com> writes:
>
> |> The fact that mathematical objects have representations composed of sets does
> |> not make them sets. It does, however, say something about the nature of sets.
>
> Hey! Very nice summary of the situation!
>
> And what it says about sets is why we love and cherish them so well.
> They were amazingly, unexpectedly, and unutterably CONVENIENT as a grand
> unifying device for all previously semi-disconnected branches of math.
>
> And, modulo complaints from categorists, have continued to be so and look
> to remain so for the forseeable future.
>
>
> |> Compare topological model theory and mereotopology--or, pointless topology--
>
> OUCH! That was an unfortunate near-compelling mistaken-uptake!
> But yes, point-free topology does sound like a plausible good idea.
>
>
> But Mitch - if you want a lot of people to understand that it ISN'T actually
> *pointless*, you had better start off much more simply than you have been doing
> so far. Just regard us as all rather dumb but vacuously willing to consider
> a new idea if we're led to it by the nose with firm hand-holding!
The presumption here is that the Boyz are open to new ideas, as long
as they can be expressed in the language of first-order logic with
identity and its set-theoretic extensions. But this is not so. For
the Boyz will brook no challenge to in-house set theories
(ZFC, NBG, etc.) or FOL= (the logic these presuppose).
If you don't believe me, ask Bill Taylor or David Ullrich or Mike Oliver
or Torkel Franzen or Robin Chapman or Daryl McCullough or George Greene
or Aatu Koskenkilta whether the orthodox extensionality principle N4
N4 AxAy[Az(z in x <-> z in y) -> x = y],
is not ruled out by (C3,C4)[Note 1];
C3 EyAx[x in y <-> Et(x in t) & P(x)] (with y not free in P(x))
C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> x = y}]
and consequently, whether N4 does not hold for classes if (C3,C4)
do.
In the absence of an answer, you might then ask any of the above
whether, in their opinion, truth in set theory or logic amounts
to anything more than coherence with received doctrine.
Regards,
John
Notes
1. The claim is:
FOL |- [((C3 & C4) -> ~N4) & (N4 -> ~(C3 & C4))]
Leonard Blackburn
>
> I am afraid that all of the mumbo-jumbo below misses the whole point
> (besides being just silly.)
I too cannot make sense of the text you quote below, but that I have
no proof that it is nonsense. What if it is my own failing for not
being able to understand?
> The concept of a set is very primitive
> and universal. It doesn't take 3 pages to explain it!
Yes, in fact, no one has yet published a correct and complete definition of
"set" and I doubt you could give one in less than 3 pages.
> It is a
> manifestation of a simple, general concept or two. And think about
> it. The idea of a set - is this a complex matter?
Extremely.
> Of course not. It
> is not complexity that makes it difficult for many people to define.
> It is actually the opposite. The concept of a set is so simple and
> primitive, that people have a hard time just formalizing what it means
> intuitively.
"formalizing what it means intuitively"? What is that supposed to mean?
Sounds like mumbo jumbo to me.
Why don't you grace us with your formal definition of set.
>
> (If a set were really as complex as you make it sound, it wouldn't be
> used throughout Mathematics. It would only be used in those
> infrequent cases where all of its various properties/requirements
> occurred.)
>
> The fact is, everything in Mathematics begins with the natural numbers
> and {true,false}. The rest is just taking various combinations of
> these primitive concepts.
The first sentence in the above paragraph is blatantly false. The
second sentence is meaningless for me since you didn't define
"various combinations".
> A set is just a mapping to {true,false},
What is a mapping? Traditionally, a set is the primitive concept and
mappings are defined in terms of sets. Therefore, your definition is
circular.
> either N=>{true,false} (N being the primitive recursively emumerable -
> i.e. representable in the system - concept) or some combination of N
> and {true,false} mapped to {true,false}.
Again, what is a "combination" in this context. Also, the part in
parentheses looks like mumbo jumbo to me. Do you mean that N is the
set of natural numbers?
In any case, what you have above is not nearly a definition of set.
It is woefully incomplete and vague. I find this ironic since you
claimed in another thread that "recursive function" has never been
given a formal definition in a publication (which is utterly false).
>
> How you interpret it is up to you: as a set, a predicate, or even as a
> single real number (each bit (instead of digit) being defined by
Strange. If every set could be interpreted as a real number then
there would be only aleph_1 sets possible. Therefore this is false,
since there are many more sets than that.
> whether or not a given value is in the set.) But all it really
> represents is a mapping (primitive or otherwise) to {true,false}.
> Beyond that you are just further manipulating these parts that make up
> the definition.
>
> Got it? Good.
Nope. I don't got anything. Your whole post was meaningless.
I challenge you to prove otherwise by giving us an actual definition
of "set" as mathematicians use the word.
John
> But careful! that doesn't mean the *conclusion* of the
> proof is nonmonotonic knowledge. It *is* imaginable that
> new information will cast doubt on the soundness of the
> theory, and there *is* a rational response to that.
Don't (C3,C4) rule out the *soundness* of theories with N4?
C3 EyAx[x in y <-> Et(x in t) & P(x)] (with y not free in P(x))
C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> x = y}]
N4 AxAy[Az(z in x <-> z in y) -> x = y]
Mightn't a rational response to *that* be a quest for bona fides?
Or, by "imaginable" did you just mean "imaginable in principle"?
And by "rational" did you just mean *rational in principle*?
And by "non-monotonic" did you just mean, "non-monotonic in principle"?
--John
"Boyz will be Boyz"
--A Nony Mous
John
<irrelevancies snipped>
> I cannot throw away ZFC ...
> If I do, then I throw away
> any chance at getting a job.
<irrelevancies snipped>
Barb Knox
mat...@math.canterbury.ac.nz (Bill Taylor) wrote:
> s...@sig.below (Barb Knox) writes:
> > Whether or not a particular idea is "completely coherent" is a matter of
> > logic.
[snip]
> > entity "exists" iff it has a model. This of course shoves the issue from
> > the "existence" of the entity to the "existence" of the model.
>
> Exactly. But there is still some gain in the shoving, I think.
I agree, but the gain seems more quantitative than qualitative, so it
doesn't solve the underlying philosophical problems.
> > Plato himself would argue that a material object such as your chair only
> > "exists" in the material universe by virtue of it being a reflection of
> > "the ideal chair" existing in the universe of forms.
>
> Did Plato go so far as positing "ideal chairs"? If so, he's even crazier
> than I thought. But I suspect you're misrepresenting him.
If I remember my Philosophy 101 correctly then he did indeed posit ideal
whatevers, of which material whatevers are but shadows. The parable of
the cave and all that.
[snip]
> > > cumulative type hierarchy of sets (V) is a coherent idea
>
> > A hard-core formalist could believe that just as sincerely
>
> Oh no, I don't think so. That's definitely going too far IMHO.
>
> > Saying that "there exists a coherent notion of ..." is
> > essentially a matter of logic,
>
> No I don't think so. "Having a consistent notion of..." may be a matter
> of logic; but that's not quite the same thing.
>
> "Having a coherent notion of..." involves a lot more stuff. I'm not sure
> exactly what, (no point arguing definitions), but I suggest it include
> imagery, inter-subjectivity, explanatory power, and so forth.
Well, I read "there exists a coherent notion of ..." as referring to
existence within the realm of human ideas, which is (as above) a matter of
logic and makes no further ontological claim. As such, it can be handled
within a formalist framework.
> > Lots of things that seem "ridiculous" happen to be matters of experimental
> > fact (e.g., quantum mechanics, relativity, the fact that the earth moves
>
> These are not really "experimental" facts, but "explanatory" facts.
> Experimental facts are what comes up on the dial when you do so-and-so.
> What positivists would call "observation statements". The things you
> mention above are at least one level higher in the scheme of science.
Quite right. I should have said "empirical fact" or "scientific fact".
> > The notion that "God is a mathematician" gained currency as more and more
>
> There is certainly a lot of evidence for that; but I think there is even more
> evidence for the case that god is an amoral experimenter who likes poking
> insects to see how they wriggle.
We're in a thick enough stew already; let's not stir "morals" into it!
> > A little philosophy is a dangerous thing...
>
> Maybe. But it never knocked down any towers.
Hmmmmm. Medieval Christianity, Islam, Marxism-Lenninism, ...
--
---------------------------
| BBB b \ barbara minus knox at iname stop com
| B B aa rrr b |
| BBB a a r bbb |
| B B a a r b b |
| BBB aa a r bbb |
-----------------------------
Mitchell Smith
Charlie-Boo wrote:
> Humble opinion? Ha! You're not being humble. You're trying to look
> like a big pseudointellectual!
>
If you will visit sci.math.research, you will find a post under "What distinguishes the identity
predicate among equivalence relations?" If you note the date on the original post, you will see
that it predates my posts on sci.logic. You may also note that the original post did attempt
present any extended philosophical justifications.
The post consisted mainly of a set of formal sentences presented in TeX with the expectation that
interested professionals could cut and paste the sentences into TeX formatting programs on their
personal computers where they could formulate their own opinions.
The post also includes two remarks. The first points out how topological convergence
properties--rather than extensional interpretation--define the diagonal map which normally
interprets the identity predicate.
The second does mention Kantian philosophy after clearly stating that conventional model theory is
rejected. This statement of rejection is necessary because the interpretation of my predicates
relies on Venn diagrams. In contrast to modern prejudices which are distrustful of visual aids,
the concept of synthetic a priori cognitions is founded on the belief that mathematical
foundations are grounded in mental imagery. Consequently, it provides a foundation for my formal
sentences whereas conventional methodologies do not.
In one of your own posts, you say:
"And when the combination is 'map N to {true, false}'
then we call it a "set".
When considering my own intuitions concerning the necessity of mutually exclusive logic, I came up
with the ideas outlined below. I apologize for the length. Much of it is generated because there
are no fonts for the symbol set I use to encode zero-order connectivity. In any case, I believe
you will find it interesting because references to the elements of {true, false} are eliminated.
These are research notes, so they are incomplete. But, the instructions are precise enough so
that you can construct the entire algebraic system if you desire.
As far as my intellectual ability is concerned, I am nothing more than a union bricklayer
(tuckpointer, actually). I just did not accept certain assumptions which seem to dominate the
foundations of mathematics. Understanding the problem entailed far more work than I had expected.
I apologize for any offense.
--------------------------------------------------------------
I. Introduction
A. Abstract
1. The material in this outline attempts to
characterize a superposition phenomenon
for zero-order logic. Specifically, the
sentential connectives are organized into
an algebraic system of sixteen products
whose arguments are references to themselves.
An automorphism which preserves those
products is described. That automorphism
has an orbit of two. References to truth
values are eliminated.
B. Architecture
1. The objective stated in the abstract is
realized by constructing a canonical
representation for the truth functional
behavior of zero-order sentential
connectives.
2. Truth tables do not satisfy expectations
as canonical representations.
a. Row order is not fixed in truth tables.
b. Mutual exclusion cannot be inferred from
the characterization of truth functional
behavior using truth tables.
C. Implementation
1. A symbol set that can convey the relationships
between complementation (unary negation) and
(DeMorgan)conjugation will be chosen.
2. Symbols will be associated with vectors from
{F, T}^4 using standard truth tables.
3. Symbols will be organized into a system of
truth products such that
abc !==> d (read as "abc 'maps to' d")
for connective symbols a, b, c, and d when
the vectors associated with a and c evaluate
componentwise to d under b.
a. Projection connectives are encoded with
the expectation of syntactic function
relative to formula generation in typical
formal language grammars.
b. The system of truth products retain
orientation with respect to grammatical
interpretation via the alteration
indicated by
A | B | A --> B
---|---|-----------
T | T | T
T | F | F
F | T | T
F | F | T
< | > | <A --> B>
---|---|-----------
T | T | T
T | F | F
F | T | T
F | F | T
in which the dual role played by the symbols
'<' and '>' facilitates interpretation of the
symbols such as '-->' as zero-order sentential
connectives.
c. Since the identification of symbols with
vectors does not involve sentence letters
or principal subformulas, assignments are
indicated with tables of the form
< | > | >--
---|---|-----------
T | T | F
T | F | T
F | T | F
F | F | F
Note that this table interprets the mapping
< >-- > !==> >--
relative to an orientation given by the
4-vectors
T T
T F
F T
F F
As an algebraic system, however, the truth
products have no dependency on the vectors
which facilitate their interpretation as
sentence connectives in zero-order logic.
D. Strategy
1. Once the system of truth products has been
constructed it becomes clear that truth functional
behavior is undetermined with respect to
conjugation.
a. If
a^* denotes the conjugate of a
and
abc !==> d
then
a^*b^*c^* !==> d^*
2. Superposition of functional states is inferred
with respect to the entire system of truth
products taken as an algebraic system over which
conjugation is an automorphism of orbit two that
preserves products.
E. Explanation
1. This property arises from the fact that the
projection connectives and their complements are
fixed under conjugation.
a. Using standard parentheses for clarity,
((A > B) <---> -(-A > -B))
((A < B) <---> -(-A < -B))
II. Symbol Set
A. General Observations
1. The symbol set is comprised of component
symbols which are reflected and rotated.
Consequently, there are no electronic fonts
available which capture the entire symbol
set.
2. One set of symbol components consists of
darts--or, angle brackets--directed to the
left, right, up, and down.
a. The left and right darts have been used
as the projection connectives in the
introductory examples.
3. The remaining symbol components consists
of line segments oriented horizontally and
vertically.
a. The line segments do not occur in the
context of sentential connectives.
b. The line segments are available for the
construction of a specific first-order
language which is not discussed in the
main points of this outline.
1'. The language construction strategy for
this particular language utilizes two
circularly defined predicates, thereby
explicitly providing for the possibility
of independent interpretation.
2'. The language construction strategy for
this particular language takes the predicate
symbols introduced using circular reference
for constructing the vectors used to interpret
truth products as sentential connectives.
a'. The horizontal and vertical line
segments may be used for the predicate
symbols of this language.
3'. The language construction strategy for
this particular language admits only one
circularly defined predicate with a syntax
that can be understood as self-defining.
a'. The self-defining primitive is taken
as the preferred truth value which
governs detachment via modus ponens
in the deductive calculus.
b'. By virtue of the mutual exclusion
imposed by this language construction
strategy, the remaining predicate
symbol is necessarily interpreted as
falsity.
4. The horizontal symbols can be constructed and used
to establish a naming convention.
a. (>, right horizontal arrowtip)
b. (<, left horizontal arrowtip)
c. (-->, right horizontal arrow-with-head)
d. (<--, left horizontal arrow-with-head)
e. (>--, right horizontal arrow-with-tail)
f. (--<, left horizontal arrow-with-tail)
g. (<-->, bidirected horizontal arrow-with-head)
h. (>--<, bidirected horizontal arrow-with-tail)
5. All horizontal symbols map to vertical symbols via
a 90 degree counterclockwise rotation
a. right horizontal arrowtip
rotates into
up vertical arrowtip
b. left horizontal arrowtip
rotates into
down vertical arrowtip
c. right horizontal arrow-with-head
rotates into
up vertical arrow-with-head
d. left horizontal arrow-with-head
rotates into
down vertical arrow-with-head
e. right horizontal arrow-with-tail
rotates into
up vertical arrow-with-tail
f. left horizontal arrow-with-tail
rotates into
down vertical arrow-with-tail
g. bidirected horizontal arrow-with-head
rotates into
bidirected vertical arrow-with-head
h. bidirected horizontal arrow-with-tail
rotates into
bidirected vertical arrow-with-tail
6. In general, complementation is implemented by
replacing an arrow-with-head by the similarly
directed arrow-with-tail and vice versa.
a. Examples
1'. '>--' is the complement of '-->'
2'. '<--->' is the complement of '>---<'
b. Exceptions
1'. Complementation between arrowtips
depends on character and line
concatenation of the language.
a'. Standard first-order grammars have
the up vertical arrowtip as the
complement of the left horizontal
arrowtip.
b'. Standard first-order grammars have
the down vertical arrowtip as the
complement of the right horizontal
arrowtip.
6. In general, conjugation is implemented by
replacing an arrow-with-head by the oppositely
directed arrow-with-tail and vice versa.
a. Examples
1'. '<--' is the conjugate of '>--'
2'. '--<' is the conjugate of '-->'
b. Exceptions
1'. Conjugation between arrowtips is governed
by the fact that projection connectives
and their complements are self-conjugate.
2'. Bidirected symbols are presumed to be
replaced by oppositely directed symbols,
although symmetry obscures this operation.
a'. '>---<' is the conjugate of '<--->'
7. Bidirected symbols are related to unidirected symbols
through logical operations preserving arrow logic.
a. Constructions to be interpreted with respect
to standard logic
1'. '<--' and '-->' if and only if '<--->'
2'. up vertical arrow-with-tail (inclusive or)
or
down vertical arrow-with-tail (nand)
if and only if
bidirected vertical arrow-with-tail (honesty)
8. Unary connectivity is implemented using elements of
the parenthesis language generated from the up vertical
arrowtip and the down vertical arrowtip.
a. The unary negation symbol is extraneous and is not
part of the formal language.
b. Unary connectivity is understood in terms of
quantifier scope.
1'. A well-construed formula evaluates to a
legitimate truth value.
2'. Well-formed formulas are not necessarily
well-construed.
3'. Satisfaction of unbound variable terms is
irrelevant since well-formed formulas which
are not well-construed may not be used as
premises for a derivation.
a'. A preferred truth value arises only
from the need to implement a rule of
detachment in the deductive calculus.
c. The syntactic elements generated from the vertical
arrowtips serve as quantifiers when immediately
prefixed to the opening delimiter of a well-formed
formula and serve as variable terms otherwise.
Mitchell Smith
John wrote:
No one can throw away Zermelo-Fraenkel set theory. It is a theory of object reference.
The entry point for recognizing this lies with understanding that mathematics has not sufficiently distinguished the
identity predicate from other equivalence relations.
This problem is reflected in the multiplicity of models inferred from the Generic Model Theorem. As with any
equivalence relation, any element of some equivalence class is a representative of the equivalence class. Ambiguous
multiplicity leads to ambiguous reference.
The resolution is to differentiate the identity predicate from the general description of equivalence. Unfortunately,
the developers of formal systems assumed that the well-established usage of equivalence relations reflected a rigorous
formulation of the identity predicate. They formulated "axioms of identity" which should actually be called "axioms of
equivalence" because these axioms are only local to a given model.
In contrast, Zermelo-Fraenkel set theory requires a notion of identity that cannot be violated by any local
formulations of equivalence. This is because such local formulations are interpreted with respect to model domains
homeomorphic to an element of the class universe. Consequently, the usual characterization of equivalence is not
sufficiently robust for Zermelo-Fraenkel set theory.
The well-established usage adopted in formal systems clearly does not constitute a rigorous formulation. Quine
acknowledges the problem in "Set Theory and Its Logic,"
"The sense of 'x=y' given by the plan of definition illustrated
in (i) and (ii) may or may not really be identity;"
"...; so here indeed is an unfavorable case, where 'x=y' does
not come out with the sense of genuine identity."
"But, even at worst, even if we do not thus rectify the
interpretation in order to sustain our method of
defining 'x=y', still no discrepancies between it and
genuine identity can be registered in terms of the
vocabulary of the theory itself."
The problem is obsfucated in texts like "An Introduction to Mathematical Logic" by Enderton and "Model Theory" by Chang
and Keisler because they invoke the well-established usage of identity as a metalinguistic predicate in order to build
their formal notion of equivalence. The failure point of this strategy is the interpretation of the universal
quantifier for Zermelo-Fraenkel set theory.
You see, it may not be Russell's paradox which precludes reference to a set of all sets. Rather, it is my personal
(and sometimes humble) opinion that it is the manner by which the identity predicate is formulated. And, I have spent
an inordinate amount of time formalizing the intuitions needed to defend this position.
Now, how do I get someone to start asking the right questions????
:-)
mitch
Mitchell Smith
Thad Coons wrote:
> "Mitchell Smith" <mit...@enteract.com> wrote in message
> news:3D9B9FBD...@enteract.com...
>
> <snipped>
>
> I must confess that most of what you wrote was far over my head.
No one knows what I am talking about. I am looking for ways to explain it. I
have spent twenty years trying to understand why my own intuitions were skewed
from what I was taught regarding the foundations of mathematics.
It is not enough, however, to complain. If one wishes to motivate change in a
discipline reflecting a century of development, one had better be able to
present a damn good case. I have a good case, but it is incredibly convoluted.
Your response seemed to be a defense of your post. There was no need. Your
questions were *awesome*!!!!
Pick any part of my post and ask some more.
By the way, a first-order algebraic structure is composed of a collection of
objects serving as a universe of discourse; a collection of symbols used in
formula formation; a collection of constant symbols, function symbols, and
predicate symbols; and an interpretation mapping that makes certain assignments.
The assignments made under the interpretation mapping constitute the extensional
interpretation of the constant symbols, function symbols, and predicate symbols.
Constant symbols are mapped directly to individual elements of the universe of
discourse. Function symbols are mapped to collections of ordered tuples--for
example, a function of the form "F(x)=y" is mapped to a collection of ordered
pairs while a function of the form "G(x, y)=z" is mapped to a collection of
ordered triples. Mappings interpreting function symbols have to respect certain
constraints that reflect the usual understanding of a function in mathematics.
Predicate symbols are also mapped to to collections of ordered tuples--ordered
pairs in the case of binary predicates. These mappings reflect truth
functionality in a mutually exclusive logic. That is, if the language terms
used to construct an atomic formula resolve to a tuple included in the
collection interpreting the predicate, the atomic formula is said to be
satisfied (Naively speaking, this means true. It is just that truth is a little
tricky in "first-order" grammars).
All of this is well and good except for the identity predicate. Because the
community of foundational mathematicians has a preference for building the
language prior to interpreting the predicates, they include the symbol for the
identity predicate among the formula generating symbols and call it a 'logical'
symbol of the language.
Well, damn it...it is a predicate and should be treated as such!
You see, it is a simple matter to extensionally interpret the identity predicate
with the collection of ordered pairs
{(x, x)| x is an element of the universe}
But, then, the symbol for the identity predicate might lose its
"well-established" meaning under a non-standard interpretation. So, instead,
they opt for "axioms of identity" which ensure that the well-established usage
of the identity predicate cannot be violated by some manipulation of the other
language symbols.
This, too, is well and good... until it fails in Zermelo-Fraenkel set theory.
I have spent a long time thinking about that failure.
Anyway, you should reread the description above. Think about most of your
regular mathematics courses and texts. Didn't your professors use definitions
most of the time? When you did your proofs, didn't you have to review
definitions just to make sure that what you wanted to assert followed from the
definitions?
The extensional interpretation of language symbols does not reflect the way
mathematics is normally conducted. Is it any wonder, then, that
Zermelo-Fraenkel set theory appears to have weaknesses? The methodology
designed to provide a formulation for the foundation of mathematics would fail
the simplest use case analysis of how mathematicians use language.
Anyway, just ask more questions if anything piqued your curiosity.
:-)
mitch
Bill Taylor
|> > evidence for the case that god is an amoral experimenter who likes poking
|> > insects to see how they wriggle.
|> We're in a thick enough stew already; let's not stir "morals" into it!
Exactly! That's why I forced them out with "amoral" (not "immoral").
|> > > A little philosophy is a dangerous thing...
|> > Maybe. But it never knocked down any towers.
|> Hmmmmm. Medieval Christianity, Islam, Marxism-Lenninism, ...
HEY! Peeeeeeeeeeeeeeep! Thank you for playing. :)
Those things are religions, not philosophy. Sure, there may be a little
philosophy strirred into a religion, just like there may be some good
health practices, but that doesn't make them the same. Religions have
always been prone to mass murder and telling lies for god.
Marxism-Leninism is no different. Like the Catholic church, they had an index,
an inquisition, veneration of saints, even holy relics! (Lenin's tomb)
Nazism was the same.
But note that Platonism, empiricism, existentialism, logical-positivism, don't
have *any* of those things, whatever other stupidities they may embrace.
So though I have no special desire to defend philosophy, let's try to keep
our conversation as clean as possible. Religions are not philosophies.
------------------------------------------------------------------------------
Bill Taylor W.Ta...@math.canterbury.ac.nz
------------------------------------------------------------------------------
Buddhism: The religion of obscurity.
Christianity: The religion of guilt.
Hinduism: The religion of backwardness.
Judaism: The religion of exclusiveness.
Islam: The religion of violence.
------------------------------------------------------------------------------
Bill Taylor
|> > Do you see a sharp line beteen the abstract and the *physical* ?
|> Not clearly.
As I suspected. Thanks for the answer.
|> "You keep using that word -- I do not think it means what you think it means"
...
|> Well, I don't know why you would have trouble conceiving the fiery chariot
|> thing. In fact it seems to me that you just did so.
Oh well - come on now. If you take that line the word becomes meaningless...
it automatically applies to *whatever* you choose to apply it to! There has
to be some more restriction than what you imply, or we're wasting our time
talking with it.
It's true I can *picture* or *imagine* the fiery chariot thing, just as I can
*picture* someone finding (or claiming to find) a contradiction in PA or
whatever. I can *picture* myself finding one, (which I would take to be
evidence of a mere mistake I hadn't found).
If all that is what you mean by "conceive of", then OK. But that seems silly.
I would think, to give the word "conceive" some more meaning than mere
childlike imagination, we need to include something to do with fitting it into
a coherent view of the world, or suchlike. But if we merely disagree on English
usage, then that's it.
|> *conceive* of that, even if you believe for abstract reasons
|> that it's "logically impossible". Do you agree that we would
|> still have rational courses of action,
I can *picture* such an alleged proof existing, and the whole math world not
being able to find a flaw in it for 10 years or more, and my still not jumping
out a window but continuing doing math. I can write myself into that story...
is that all you mean?
|> and indeed that we would still be able to use (some) mathematics?
That's more doubtful. It doesn't usually take more than a year or so for
the math world to sort out an apparent paradox with preliminary comfort.
I could conceive maybe that they managed to pin down the proof onto some
dodgy concept such as impredicativity, but nothing simpler than that.
But every time this sort of topic comes up, Mike, I ask you in return,
how would you react if someone managed to prove that 34 + 53 =/= 87.
By a means you could follow and agree with. What would be your response?
And you always reply, something to the effect that there are degrees of
unbelievability, or whatever. But your replies strike me as very evasive;
that's not an accusation - I know you don't intend to be evasive, but that's
the way it comes across.
Is it conceivable to you that 5 + 8 = 15 ? 2 + 3 = 7 ?
What's the difference between that and some of the above? (No need to reply
to this - merely rhetorical - the answer will be as above I'm sure.)
------------------------------------------------------------------------------
Bill Taylor W.Ta...@math.canterbury.ac.nz
------------------------------------------------------------------------------
The more the universe seems comprehensible,
the more it also seems pointless.
------------------------------------------------------------------------------
Bill Taylor
|> My point is that induction is about second-order objects (properties,
|> rather than numbers).
It doesn't have to be. We can always fall back on the old schema approach.
You will doubtless call that bringing back poperties by the back door,
but I'm not so convinced, (nor is the orthodox math world, seemingly).
i.e. we don't have to worry about *all propoerties* but just those which we
can write down. I suspect this makes a huge philosophical difference.
But maybe I'm just bleating.
Or maybe you were.
Let someone else decide.
------------------------------------------------------------------------------
Bill Taylor W.Ta...@math.canterbury.ac.nz
------------------------------------------------------------------------------
Civilization is a slow process of adopting the ideas of minorities
------------------------------------------------------------------------------
Mike Oliver
>|> that it's "logically impossible". Do you agree that we would
>|> still have rational courses of action,
>
> I can *picture* such an alleged proof existing, and the whole math world not
> being able to find a flaw in it for 10 years or more, and my still not jumping
> out a window but continuing doing math. I can write myself into that story...
> is that all you mean?
Well, write yourself into a bit stranger story. It's not just a proof
in which the math world can't find an error. It's a formal derivation
of tractable length -- let's say, 10 pages or so of ordinary print.
Every step has been checked line-by-line, over and over again thousands
of times, by man and machine.
Now continue the story. How do you respond? You could:
A) take the attitude that there is some nonzero probability, each time
a line is checked, of missing an error, and therefore no matter how
many times each line has been checked without finding it, it has
simply been missed, and the *next* check will probably find it.
B) decide that logic, or at least mathematics, has now been refuted
and is useless -- we may as well go studying the grimoires, because
magic is loose in the world. Or *we're* just nuts and the nice lady
will come in with our dinner soon.
C) decide that mathematics has worked pretty well up to now and
there's no reason to expect it to stop; we'll just have to revise
our confidence as to how far it can be extrapolated, and also come
up with a different story about why (say) calculus is still reliable
for building bridges.
D) something else -- you fill it in.
My analysis is as follows: (A) is reasonable for a while but past
a certain point becomes stupid bullheadedness, refusing to see what's
in front of you. (B) is silly defeatism; (C) is the correct and
rational response.
> But every time this sort of topic comes up, Mike, I ask you in return,
> how would you react if someone managed to prove that 34 + 53 =/= 87.
> By a means you could follow and agree with.
It would have to be an extremely primitive "means" in order for me
not to simply decide that I was wrong to agree with the "means", of
course. But let's suppose it was. It's hard to imagine what
such a "means" could be -- I don't know, let's say if I put 3 matches
and 5 matches facing East-West and then turned the table North-South
I now had only 2 matches, even though I was watching them all and none
disappeared, and this happened over and over and over again, no matter
how many times I tried it.
Well, then I guess I'd have to choose (B). That's the distinction,
for me, between the "really finite" and the abstractly finite --
whether you choose (B) or (C).
Mitchell Smith
John wrote:
> For
> the Boyz will brook no challenge to in-house set theories
> (ZFC, NBG, etc.) or FOL= (the logic these presuppose).
Every model of Zermelo-Fraenkel set theory is a model of the definitions and axioms I have presented in
whichever posts they might occur. My work is not a challenge to standard mathematics in any way. Rather, it
is a challenge to the metamathematics that leaves the language symbols of Zermelo-Fraenkel set theory without
interpretation relative to the class universe.
>
> If you don't believe me, ask Bill Taylor or David Ullrich or Mike Oliver
> or Torkel Franzen or Robin Chapman or Daryl McCullough or George Greene
> or Aatu Koskenkilta
I certainly owe Aatu an apology for an evening when I was a little tired. That has been given indirectly in
another post and reasserted here directly.
(I am sorry for the character of my comments in that post. Your answers were knowledgeable and forthright.)
> whether the orthodox extensionality principle N4
>
> N4 AxAy[Az(z in x <-> z in y) -> x = y],
>
> is not ruled out by (C3,C4)[Note 1];
>
> C3 EyAx[x in y <-> Et(x in t) & P(x)] (with y not free in P(x))
> C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> x = y}]
>
I assume
C3 EyAx[x in y <-> {Et(x in t) & P(x)}] (with y not free in P(x))
because my precedence rules are a little rusty.
I also assume that P(x) represents a formula from the language with some indeterminate set of language
symbols.
So, C3 is saying that every formula free for x that is absolute for a given model has an associated class
consisting of elements from the model. But, this is true by definition of absoluteness:
Let M be a transitive class and let phi(x_1, ..., x_n) be a
formula. We say that phi is absolute for M if
M satisfies phi(x_1, ..., x_n) iff phi(x_1, ..., x_n)
for all x_1, ..., x_n in M.
Perhaps, however, C3 is somewhat more restrictive in that it asserts that the class must be an element of the
model. That must be why it is so similar to the form for restricted separation:
Ap_1 ... Ap_nAtEyAx[x in y <-> {x in t & P(x, p_1,...,p_n)}] ,
where P is a restricted formula.
The relevant corollary to the Normal Form Theorem states:
If M is a Goedel closed transitive class, then M satisfies
Restricted Separation
The phrase "M satisfies Restricted Separation" means that the form for restricted separation is satisfied for
every restricted formula P.
So, while I do not have time to contemplate the full implications of C3 in a general context, it appears that
the underlying intuition is in good company. Absoluteness and restricted separation are both essential to
describing the constructible universe as a model for Zermelo-Fraenkel set theory.
Now, C4 seems reasonable. It is attempting to assert that if two term referents are equivalent with respect
to extensionality, then there is a class associated with those references that is an atom of the class
universe understood as a partial order. Naturally, that is a quick paraphrase assuming much more than is
indicated.
But, what of its contrapositive? You could have... No, you are probably assuming those ludicrous axioms of
identity. So, if x=y your premise is true whence your atom must be there. If not(x=y), the axiom of pairing
gives your existence statement and extensionality fails.
It is expected that I assume pairing, is it not? I'm flying by the seat of my pants here.
Am I missing something? It appears to me that N4 --> C4. That is, if the premise of N4 is false, then the
premise of C4 is false. And, if the consequent of N4 is true, the applicable axiom of identity forces your
atom to exist as noted above, whence the consequent of C4 is true. So, when N4 is true, C4 is true.
That means you are claiming that N4 --> not(C3). This might actually occur, although I do not see it
immediately. However, since it appears that your underlying intuition is related to issues surrounding the
construction of Goedel's constructible universe, it is probably something that you would not want anyway. Both
the axiom of extensionality and restricted separation are satisfied by the constructible universe.
>
> and consequently, whether N4 does not hold for classes if (C3,C4)
> do.
>
> In the absence of an answer, you might then ask any of the above
> whether, in their opinion, truth in set theory or logic amounts
> to anything more than coherence with received doctrine.
>
> Regards,
> John
>
> Notes
>
> 1. The claim is:
>
> FOL |- [((C3 & C4) -> ~N4) & (N4 -> ~(C3 & C4))]
Drop the FOL. If the membership predicate is being used in the formal language of a first-order grammar with
a set-theoretic context, it is not FOL with its generic language elements. Try the usual MFST (my favorite
set theory).
Also, if you wish to respond to this, you may consider moving it to a different post. The material here could
lead to issues not germane to fundamental intuitions concerning the nature of sets (And, I say that with the
full realization that nobody is understanding my posts.).
I would be interested in any explanation on how C3 differs from restricted separation. I do not have time to
play with the quantifiers to see if it is logically equivalent. And, of course, do you have a model?
:-)
mitch
R. Srinivasan
> and {true,false}. The rest is just taking various combinations of
> these primitive concepts. A set is just a mapping to {true,false},
> either N=>{true,false} (N being the primitive recursively emumerable -
> i.e. representable in the system - concept) or some combination of N
> and {true,false} mapped to {true,false}.
>
> How you interpret it is up to you: as a set, a predicate, or even as a
> single real number (each bit (instead of digit) being defined by
> whether or not a given value is in the set.) But all it really
> represents is a mapping (primitive or otherwise) to {true,false}.
> Beyond that you are just further manipulating these parts that make up
> the definition.
>
I'm sorry Mr. Charlie-Boo,
But this is just plain Charlie-Poo.
A set is an undefined concept in set theory, period. You can interpret
a finite set as a "collection of things" and give it intuitive
meaning, but not an infinite set. What does "all" mean in the set of
all natural numbers (N)? The existence of non-standard models for N
shows that "all" cannot be understood in any simple, intuitive manner
(if at all it can be understood).
Sincerely,
R. Srinivasan srad...@in.ibm.com
Thad Coons
> No one knows what I am talking about. I am looking for ways to explain
it. I
> have spent twenty years trying to understand why my own intuitions were
skewed
> from what I was taught regarding the foundations of mathematics.
>
> It is not enough, however, to complain. If one wishes to motivate change
in a
> discipline reflecting a century of development, one had better be able to
> present a damn good case. I have a good case, but it is incredibly
convoluted.
>
> Your response seemed to be a defense of your post. There was no need.
Your
> questions were *awesome*!!!!
I while ago, I looked at Euclid's elements. By modern standards, his
definitions are vague and imprecise, and there are a lot of ambiguities and
unconscious assumptions that have arisen and been resolved over the
centuries. Mathematical foundations is a much newer discipline, but a
century of work with it ought to let us take a better look at concepts that
the founders of the discipline may have grasped only dimly.
> By the way, a first-order algebraic structure is composed of a collection
of
> objects serving as a universe of discourse; a collection of symbols used
in
> formula formation; a collection of constant symbols, function symbols, and
> predicate symbols; and an interpretation mapping that makes certain
assignments.
I can understand that. One of the problems with set theory is that when
sets are used to describe a universe of discourse, they are generally taken
to be ordinary (non-self-inclusive). While this approach can be used without
problem to describe numbers, points, functions, and so forth, it presents
real difficulties in trying to describe set theory itself. When the universe
of discourse its itself a set, the theory instantly becomes
self-referential, which introduces the possiblity of Russell's paradox. The
familiar concepts that work well in other areas of mathematics break down,
because there is no such animal as an ordinary set of all, and only,
ordinary sets.
If the paradox is taken as the theorem of logic it is, it is possible to
conclude from that every set has to either exclude least one ordinary set
(namely, itself) or include at least one self-inclusive and non-ordinary
set. (namely itself). That's the exclusive or, BTW.
It doesn't matter what kind of collection you use or what form of
membership or inclusion takes, either. As soon as the predicate becomes
well defined, the same dichotomy is enforced. Mereology and the subset
relation take the self-inclusive option, classical set theory takes the self
exclusive one. Attempts to evade the paradox using non-standard logic or
some kind of fuzzy set theory aren't promising, either.
Thus it appears that we can we can confidently speak of ANY ordinary
set, but the concept of ALL of them is utterly and irremediably intractable
and self-contradictory. Set theory works so well in other areas, that the
failure of intuition and standard techniques when it is used to examine
itelf gave mathematicians absolute fits, and still does. People still tie
themselves in knots trying to define and understand the set-theoretical
universe. Russell's theory of types, ZF and Von Neuman are the best
recognized attempts, but the ghost of paradox lurks everywhere, such as when
ZF is used incautiously as the basis of number theory. It must be banished
by elaborate incantations (or, so it appears to the uninitiated)
Umm-- what IS the identity predicate?
> You see, it is a simple matter to extensionally interpret the identity
predicate
> with the collection of ordered pairs
>
> {(x, x)| x is an element of the universe}
>
> But, then, the symbol for the identity predicate might lose its
> "well-established" meaning under a non-standard interpretation. So,
instead,
> they opt for "axioms of identity" which ensure that the well-established
usage
> of the identity predicate cannot be violated by some manipulation of the
other
> language symbols.
>
> This, too, is well and good... until it fails in Zermelo-Fraenkel set
theory.
> I have spent a long time thinking about that failure.
The failure you speak of doesn't sound familiar. Could you elaborate?
Thad Coons
Aatu Koskensilta
Bill Taylor wrote:
> Aatu Koskensilta <aatu.kos...@xortec.fi> writes:
>
> |> As an example, Gerog Kreisel has found a proposition p, of which the
> |> shortest proof in FOL has more steps than the visible universe has
> |> particles, but which has 5 step proof in second order logic.
>
> Aatu - can you expand on this a bit please? I've dimly heard of such
> a thing, but forgotten what it was. Can you please outline Kreisel's
> (or any other) example for us?
I'll do some digging this week-end and come back to this next week; I'm
not sure enough of the details of the Kreisel construction to give an
exposition without a text to fall back onto :)
> --------
>
> Regarding the other thread, we seem to agree pretty much on the scope of
> a proper view of formalism/finitism; but there's just one point of disagreement
> which didn't seem important enough for a separate reply.
> You say that a commitment to the meaningfulness of decidable statements
> does NOT entail a commitment to a realist view of natural numbers. As you note,
> I disgree, but I suspect we're disagreeing over mere philosophical definitions,
> and not anything serious. So I doubt there's any need to continue on this one
> - it is not a math-philosophical disagreement, merely a linguistic one, I suspect.
Most likely the disagreement is over terminology, yes.
> Cheers! Nice to see you joining the group!
Thanks. I've been around for some time, I usually just don't post that much.
--
Aatu Koskensilta (aatu.kos...@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
John
> Aatu Koskensilta <aatu.kos...@xortec.fi> writes:
>
> |> As an example, Gerog Kreisel has found a proposition p, of which the
> |> shortest proof in FOL has more steps than the visible universe has
> |> particles, but which has 5 step proof in second order logic.
>
> Aatu - can you expand on this a bit please? I've dimly heard of such
> a thing, but forgotten what it was. Can you please outline Kreisel's
> (or any other) example for us?
>
>
> Regarding the other thread, we seem to agree pretty much on the scope of
> a proper view of formalism/finitism; but there's just one point of disagreement
> which didn't seem important enough for a separate reply.
>
> You say that a commitment to the meaningfulness of decidable statements
> does NOT entail a commitment to a realist view of natural numbers. As you note,
> I disgree, but I suspect we're disagreeing over mere philosophical definitions,
> and not anything serious. So I doubt there's any need to continue on this one
> - it is not a math-philosophical disagreement, merely a linguistic one, I suspect.
>
>
There was a gonflé named Gaylord
Whose rants were crafted and taylored
To ignore or abuse
Unfashionable views,
And logics that Gaylord judged wayward.
--John
Bill Taylor
|> In simplest terms, the foundations of mathematics should be based upon defined
|> language elements.
This sounds very dubious. Sure they can be *explicated* by natural language,
they have to be. But to talk about defining them that way sounds dangerous -
defining the more precise in terms of the less precise is a dodgy way to go.
|> I infer from all of the literature that everyone believes Russell's paradox
|> to be a condition of reality.
That may be the literature, but that is probably mostly philosophical (rather
than math-logical) and thus likely to be rubbish.
On this newsgroup, the converging opinion of the "intelligentsia" here :)
seems to be that it is a fairly trivial matter that should never have arisen
in the first place, and is easily and properly disposed of by noting its
ill-foundedness of reference. Most of the other paradoxes are ditto.
This view does not sit well with those who like to be confused or argue endlessly.
|> I view it as a reflection of illiteracy.
Not sure what you mean by this, but if you want to use harsh words, just call it
a reflection of stupidity. More accurate.
|> If someone will ask questions in small pieces,
No can do yet. We must have something to question about. Can you not write
a brief summary of your views in plain English? Or if not, a summary ditto
of the sort of thing that has been nagging at your mind for 20 years.
I regret to say you have not been very coherent as yet. But I remain hopeful.
------------------------------------------------------------------------------
Bill Taylor W.Ta...@math.canterbury.ac.nz
------------------------------------------------------------------------------
That man is dangerous who only cares about one thing
------------------------------------------------------------------------------
Mitchell Smith
Bill Taylor wrote:
> Mitchell Smith <mit...@enteract.com> writes:
>
> |> In simplest terms, the foundations of mathematics should be based upon defined
> |> language elements.
>
> This sounds very dubious. Sure they can be *explicated* by natural language,
> they have to be. But to talk about defining them that way sounds dangerous -
> defining the more precise in terms of the less precise is a dodgy way to go.
>
Both Carnap and Ramsey investigated the possibility of using circular reference in
their definitions. Both acknowledged that there was nothing inherently paradoxical in
such constructions.
Definitions directed at introducing constants or functions in such a manner are
ill-advised because it does not make sense to say that the description (definiendum)
of an object is well defined if and only if it is well defined.
Predicates, however, are a different story. A predicate need only reflect a useful
property. This can be done through definition alone.
>
> |> I infer from all of the literature that everyone believes Russell's paradox
> |> to be a condition of reality.
>
> That may be the literature, but that is probably mostly philosophical (rather
> than math-logical) and thus likely to be rubbish.
>
> On this newsgroup, the converging opinion of the "intelligentsia" here :)
> seems to be that it is a fairly trivial matter that should never have arisen
> in the first place, and is easily and properly disposed of by noting its
> ill-foundedness of reference. Most of the other paradoxes are ditto.
>
> This view does not sit well with those who like to be confused or argue endlessly.
>
Noting its ill-foundedness of reference constitutes "belief in Russell's paradox as a
condition of reality."
To be more precise, however, it is not the paradox itself which presents an issue. It
is the inference made from the paradox which cannot be justified.
>
> |> I view it as a reflection of illiteracy.
>
> Not sure what you mean by this, but if you want to use harsh words, just call it
> a reflection of stupidity. More accurate.
>
Illliteracy is the correct word when suggesting that language usage is the underlying
problem.
There is a simple belief that the well-established usage of equivalence relations
constitutes a rigorous formulation of the identity predicate. It does not. The
problem presents itself when attempting to interpret the universal quantifier for
Zermelo-Fraenkel set theory.
It it true that Russell's paradox precludes an extensional interpretation for this
symbol. However, it is possible to construct the language--by beginning with
definitions for predicates using circular reference--so that every extensional
interpretation will be constrained in specific ways. This constraint constitutes a
meaningful intensional interpretation for the symbol.
> |> If someone will ask questions in small pieces,
>
> No can do yet. We must have something to question about. Can you not write
> a brief summary of your views in plain English? Or if not, a summary ditto
> of the sort of thing that has been nagging at your mind for 20 years.
>
The language symbols of Zermelo-Fraenkel set theory have no interpretation with
respect to the term "the class universe."
Naive, briefly summarized, and math-logical approaches have not contributed to a
resolution of that problem.
>
> I regret to say you have not been very coherent as yet. But I remain hopeful.
That may reflect your expectations more than my abilities. I have been driven to
understand certain ideas of Immanuel Kant in order to understand how to interpret
circularly defined predicates. You are probably unaware that he had specific views
concerning the foundations of mathematics which are only partially captured in the
extensional interpretations which currently dominate mathematical logic. In fact,
those views are the foundation of his epistemology.
Most people who attempt to read Kant find him to be intractable, as well. Your
assessment of my statements, however, seem to be at odds with those of another person
whom I met on newsgroups since I began several weeks ago:
"Yea! I corresponded with Kant scholars for quite
some time on alt.philosophy.kant, but nowhere have
I come across such lucid explanations as you have
given me."
Unfortunately, the individual in question lacks the mathematical background to assess
a formal system for reasonableness.
:-)
mitch
Mitchell Smith
Thad Coons wrote:
>
> Umm-- what IS the identity predicate?
>
The definition for "proper_part" is given by:
for all a for all b( a proper_part b iff (
forall c( b proper_part c implies a proper_part c )
and
exists c( a proper_part c and not (b proper_part c) ) ) )
The definition for "in" is given by:
for all a for all b( a in b iff (
for all c( b proper_part c implies a in c )
and
exists c( a in c and not (b proper_part c) ) ) )
The definition for "equiv" is given by
for all a for all b( a equiv b iff (
for all c ( a proper_part c iff b proper_part c )
and
for all c ( c proper_part a iff c proper_part b )
and
for all c ( a in c iff b in c )
and
for all c ( c in a iff c in b ) )
The definition for "=" is given by
a = b iff for all c ( a in c iff b in c )
To be compatible with Zermelo-Fraenkel set theory you must have
a equiv b iff for all c ( a proper_part c iff b proper_part c )
a equiv b iff for all c ( c proper_part a iff c proper_part b )
a equiv b iff for all c ( a in c iff b in c )
a equiv b iff for all c ( c in a iff c in b )
as theorems.
This requires the axioms:
1. Axiom of Conjoinment
for all a for all b(
for all c( c proper_part a iff c proper_part b )
implies
for all c( a in c iff b in c ) )
2. Axiom of Constituency
for all a for all b(
for all c( c in a implies c in b )
implies
for all c( b proper_part c implies a proper_part c ) )
3. Axiom of Power
for all a exists b for all c(
c in b iff c proper_part a )
The identity predicate is obtained by choosing that theorem for the "equiv"
relation which conveys topological separation of points when negated.
Extensionality leads to the weakness exploited by Cohen's forcing language to
obtain the Generic Model Theorem because extensionality is related also to the
simple order which can be interpreted independently as a mereological
predicate. That is how Cohen's forcing language can do its "magic."
Mitchell Smith
Bill Taylor wrote:
> |> If someone will ask questions in small pieces,
>
> No can do yet. We must have something to question about.
The definition for "proper_part" is given by:
for all a for all b( a proper_part b iff (
forall c( b proper_part c implies a proper_part c )
and
exists c( a proper_part c and not (b proper_part c) ) ) )
The definition for "in" is given by:
for all a for all b( a in b iff (
for all c( b proper_part c implies a in c )
and
exists c( a in c and not (b proper_part c) ) ) )
The definition for "equiv" is given by
for all a for all b( a equiv b iff (
for all c ( a proper_part c iff b proper_part c )
and
for all c ( c proper_part a iff c proper_part b )
and
for all c ( a in c iff b in c )
and
for all c ( c in a iff c in b ) )
as theorems.
This requires the axioms:
1. Axiom of Conjoinment
2. Axiom of Constituency
3. Axiom of Power
Cohen's forcing language manipulates the topology on the class universe.
Failure to have agreement between subject and verb is an example of illiteracy from
natural language usage. Failing to recognize distinct contexts in formal systems is
an example of illiteracy in formal language usage.
Thad Coons
"Mitchell Smith" <mit...@enteract.com> wrote in message
news:3D9CE745...@enteract.com...
>
> You see, it may not be Russell's paradox which precludes reference to a
set of all sets. Rather, it is my personal
> (and sometimes humble) opinion that it is the manner by which the identity
predicate is formulated. And, I have spent
> an inordinate amount of time formalizing the intuitions needed to defend
this position.
Russell's paradox by itself doesn't preclude reference to a set of all sets.
What it does forbid is the neat, clean division of sets into self-inclusive
and not-self-inclusive categories. The self-inclusive kind doesn't pose a
problem, but the non-self-inclusive category contradicts its own definition.
It's the self-reference that makes this different from trying to categorize
points, numbers, and potatoes with predicates and logical functions.
Thad Coons
Mitchell Smith
Thad Coons wrote:
>
> Russell's paradox by itself doesn't preclude reference to a set of all sets.
Agreed. I wish I had your clarity.
> but the non-self-inclusive category contradicts its own definition.
Which is good reason for turning to a mereological formulation of the universal
class (by which I mean that the satisfaction predicate should be an order
relation and not the membership relation).
To implement this strategy, one must understand that the "part" predicate (a
reflexive order) is the satisfaction predicate. The type difference needed to
control reference to the class universe is implemented with an exclusive
disjunction between the mereotopological "proper_part" predicate and the
"characteristic equivalence relation for the model" which is my way of saying
the the equivalence class derived from the axioms of identity.
Now, we do not wish to do mereology. So, we want an identity predicate
formulated with respect to the membership predicate.
Extensionality, however, poses a problem because it characterizes equivalence
with the same semantics that will be used to redefine "proper_part" as
"proper_subset" and "part" as "subset." Contexts need to be managed here.
Instead, you characterize the identity predicate in terms of the topological
separation of points. Then the mereological satisfaction predicate is
independent of the problem exposed by Russell's paradox. Moreover, the
multiplicity implied by the Generic Model Theorem is collapsed into the
mereological definition of a point.
This provides an *intensional* interpretation of the universal quantifier as was
originally expected when set theory was being formalized.
The collection semantics for the class universe obtained in this way comes from
an axiom that explicitly asserts almost universality.
The details for constructing this identity predicate require one to understand
how the equivalence relation derived from the axioms of identity is not robust.
:-)
mitch
Mitchell Smith
Mitchell Smith wrote:
> John wrote:
>
> Also, if you wish to respond to this, you may consider moving it to a different post. The material here could
> lead to issues not germane to fundamental intuitions concerning the nature of sets (And, I say that with the
> full realization that nobody is understanding my posts.).
Dumb remark. Getting too tired again. Sorry. Any discussion of specific models of Zermelo-Fraenkel set theory
are--by definition--germane to intuitions concerning the nature of sets.
:-)
mitch
John
> Mitchell Smith <mit...@enteract.com> writes:
>
>
> |> I infer from all of the literature that everyone believes Russell's paradox
> |> to be a condition of reality.
>
> That may be the literature, but that is probably mostly philosophical (rather
> than math-logical) and thus likely to be rubbish.
>
> On this newsgroup, the converging opinion of the "intelligentsia" here :)
> seems to be that it is a fairly trivial matter that should never have arisen
> in the first place, and is easily and properly disposed of by noting its
> ill-foundedness of reference. Most of the other paradoxes are ditto.
Another pop for the Party Line by a resident illuminati.
-John
|> ... is an anagram of ASS JAM HIRER
|> I hope it may be possible to generalise this result.
REAR HAS JISM
=============
--------------------------
| |
| Ascii-art picture |
| deleted by university |
| auto-censoring device |
| |
--------------------------
news:<a71vt1$dbb$1...@cantuc.canterbury.ac.nz>
John
> > Russell's > paradox
> |> to be a condition of reality.
>
> That may be the literature, but that is probably mostly philosophical (rather
> than math-logical) and thus likely to be rubbish.
>
> On this newsgroup, the converging opinion of the "intelligentsia" here :)
> seems to be that it is a fairly trivial matter that should never have arisen
> in the first place, and is easily and properly disposed of by noting its
> ill-foundedness of reference. Most of the other paradoxes are ditto.
>
> This view does not sit well with those who like to be confused or argue endlessly.
>
> |> I view it as a reflection of illiteracy.
>
> Not sure what you mean by this, but if you want to use harsh words,
> just call it a reflection of stupidity. More accurate.
Interesting that Taylor doesn't cite *his own* account of Russell's
Paradox. In article news:<Bx2AL...@cantua.canterbury.ac.nz>
w...@math.canterbury.ac.nz (Bill Taylor) wrote:
> The trouble with the Russell set, (or the universal set), is this.
> Attempting to define it by naive comprehension as R = { x | phi(x) },
> is objectionable; not (though) because phi may implicitly *refer to*
> itself, which is mere "harmless impredicativity", but because the
> **range of candidate members** includes R itself. This is the key
> to "vicious circle" impredicativity, it seems to me.
> To define a *set* (by comprehension), means to consider its
> possible members, so the set itself cannot possibly be considered
> as a prospective member, because it has not (yet) been defined.
> Doubtless this will sound just like "harmless impredicativity" to
> many, but I think it is a distinguishable special case, and the
> one that contains all the venom of the paradox. To attempt to
> define a set by looking at a prospective member as possibly
> being the set itself, seems to be *asking* for trouble.
> A true vicious circle; far worse than "harmless".
> It seems to be a hopeless confounding of the very meaning
> of "membership".
Notwithstanding Taylor's eristic flatulence, Russell's Paradox has
no more to do with sets and membership than the Barber's Paradox
has to do with barbers and shaving, as it was also necessary to
explain to David Ullrich:
> Six weeks of Logic 101 would enable anyone other than a brain-dead
> analysis prof to grasp that (1) and (2) are both instances of the
> inconsistent (3):
>
> 1) EyAx(x in y <-> ~(x in x))
> 2) EyAx(x is shaved by y <-> ~(x is shaved by x))
> 3) EyAx(xMy <-> ~(xMx))
>
> The inconsistency of (1) and (2) has no more to do with sets or
> membership or shaving than the cogency of (3) has to do with
> mathies, dweebness or Dullrich.
>
> 4) All mathies are dweebs.
> Dullrich is a mathie.
> Ergo: Dullrich is a dweeb.
news:<70f94e16.0208...@posting.google.com>:
Are the 'intelligentsia' here *all* refugees from Logic 101?
--John
Mitchell Smith
Aatu Koskensilta wrote:
> They're not, they have models, namely natural numbers and
> the cumulative hierarchy.
With all due respect, the cumulative hierarchy is not a model. If it were, Goedel
would not have had any motivation to formulate the constructible universe.
The cumulative hierarchy only specifies a topological covering.
By definition,
A transfinite sequence <H_alpha: alpha in Ord> is called a
cumulative heirarchy if
(i) H_alpha subset H_alpha+1 and
H_alpha+1 subset P(H_alpha)
(ii) if alpha is a limit ordinal then
H_alpha = U_beta<alpha H_beta,
where U denotes the union operation.
For any set A, the class of all sets constructible from A is a model specified by a
cumulative heirarchy.
However, when speaking of "the cumulative heirarchy" as a model, your reference is not
well formulated. The cumulative heirarchy in its general formulation is not a model
because cardinal concepts are not absolute.
Of course, if it were possible to formulate a cumulative heirarchy specifying the
class universe, then the cardinal concepts which fail generally would be absolute for
that model. But, alas, Russell's paradox precludes you from forming such a model in
conventional model theory.
Which cardinal concepts fail? Jech lists
Y=P(X), the power set
|Y|=|X|, equinumerosity
alpha is a cardinal
beta = cf(alpha)
alpha is a regular cardinal
as expressions known not to be absolute.
:-)
mitch
John
Take Bill Taylor and company with more than a grain of salt. And when
you're tired, get some sleep.
--John
Mitchell Smith
John wrote:
> Interesting that Taylor doesn't cite *his own* account of Russell's
> Paradox. In article news:<Bx2AL...@cantua.canterbury.ac.nz>
> w...@math.canterbury.ac.nz (Bill Taylor) wrote:
>
> > The trouble with the Russell set, (or the universal set), is this.
> > Attempting to define it by naive comprehension as R = { x | phi(x) },
> > is objectionable; not (though) because phi may implicitly *refer to*
> > itself, which is mere "harmless impredicativity", but because the
> > **range of candidate members** includes R itself. This is the key
> > to "vicious circle" impredicativity, it seems to me.
>
> > To define a *set* (by comprehension), means to consider its
> > possible members, so the set itself cannot possibly be considered
> > as a prospective member, because it has not (yet) been defined.
> > Doubtless this will sound just like "harmless impredicativity" to
> > many, but I think it is a distinguishable special case, and the
> > one that contains all the venom of the paradox.
In at least one of my posts I recite:
It makes no sense to say that an object is well defined iff and only if it is well defined.
> To attempt to
> > define a set by looking at a prospective member as possibly
> > being the set itself, seems to be *asking* for trouble.
> > A true vicious circle; far worse than "harmless".
> > It seems to be a hopeless confounding of the very meaning
> > of "membership".
>
You would have to know the meaning of membership in a rigorous sense in order to decide if it was confounding
the meaning of membership. Actually, the violation has nothing to do with the membership predicate.
Everyone learns about well-defined functions in algebra. You are expected to recognize when well definition is
an issue and when it simply follows from constructive principles applied to functions known to be well-defined.
For formal systems, it is also important to understand constant bindings as well-defined in the sense that they
consistently resolve to the same individual object of the model domain.
Formal systems whose notion of truth relies on extensional interpretations of its language symbols violate this
basic principle. In simple cases, you make sure that you make assignments that *reflect* expectations with
regard to well definition. But, because you are not formulating well-defined descriptions in the language, you
open the possibility to abuses such as impredicatively defined objects.
>
> Notwithstanding Taylor's eristic flatulence, Russell's Paradox has
> no more to do with sets and membership than the Barber's Paradox
> has to do with barbers and shaving, as it was also necessary to
> explain to David Ullrich:
>
> > Six weeks of Logic 101 would enable anyone other than a brain-dead
> > analysis prof to grasp that (1) and (2) are both instances of the
> > inconsistent (3):
> >
> > 1) EyAx(x in y <-> ~(x in x))
> > 2) EyAx(x is shaved by y <-> ~(x is shaved by x))
> > 3) EyAx(xMy <-> ~(xMx))
> >
Correct.
The problem here is the self-inconsistent formula. But, do you see the solution?
You need to have a different (satisfaction) predicate for formulating the universal class. It must express
comparability with all term referents. Use the characteristic equivalence relation of the language (identity
to all the rest of you) and exclusive disjunction to enforce a type difference between the universal class and
all other term referents. Then, relate the two predicates so that the universal class is left with the
collection semantics you would have had by starting with membership as the satisfaction predicate.
Comparability and isolation--the opposite of divide and conquer.
:-)
mitch
Mitchell Smith
John wrote:
Thanks.
My exhaustion does not get resolved with sleep, unfortunately. Several years ago I was playing with syntactic
forms--truth tables actually--and started to "organize" the forms. It turned into a game that does not stop.
Normally truth tables are interpreted by rows. I started playing with columns. The problems started when I just
exchanged 'T' and 'F' for horizontal and vertical lines.
For example, there is a reflection,
|
---
|
|
***************************
|
|
---
|
and the usual complementation,
| ---
| ---
--- |
| ---
Anyway, what I did not realize was that my decision procedure for organizing these "parity grammars" was essentially
an application of the four-color map theorem.
Normally we think of the truth table as fixing a context for the interpretation of a connective symbol. We get that
from extracting propositional forms from reasoned arguments in natural language. This is just fine for a
sophisticated language user who enjoys a fixed context in the Platonistic/Cartesian/Newtonian idealism.
But, for a biological organism whose neural net presumably operates according to threshold logic, context is not
fixed. Consistent contexts like those used by analytical methods invoke a normally transparent decision procedure to
manage
(page quickly... there are 24 of them)
< | > | -->
---|---|-----------
T | T | T
T | F | F
F | T | T
F | F | T
< | > | -->
---|---|-----------
T | F | F
T | T | T
F | F | T
F | T | T
< | > | -->
---|---|-----------
T | T | T
T | F | F
F | F | T
F | T | T
< | > | -->
---|---|-----------
T | F | F
T | T | T
F | T | T
F | F | T
< | > | -->
---|---|-----------
F | T | T
F | F | T
T | T | T
T | F | F
< | > | -->
---|---|-----------
F | F | T
F | T | T
T | F | F
T | T | T
< | > | -->
---|---|-----------
F | T | T
F | F | T
T | F | F
T | T | T
< | > | -->
---|---|-----------
F | F | T
F | T | T
T | T | T
T | F | F
< | > | -->
---|---|-----------
T | T | T
F | T | T
T | F | F
F | F | T
< | > | -->
---|---|-----------
T | F | F
F | F | T
T | T | T
F | T | T
< | > | -->
---|---|-----------
T | T | T
F | F | T
T | F | F
F | T | T
< | > | -->
---|---|-----------
T | F | F
F | T | T
T | T | T
F | F | T
< | > | -->
---|---|-----------
F | T | T
T | T | T
F | F | T
T | F | F
< | > | -->
---|---|-----------
F | F | T
T | F | F
F | T | T
T | T | T
< | > | -->
---|---|-----------
F | T | T
T | F | F
F | F | T
T | T | T
< | > | -->
---|---|-----------
F | F | T
T | T | T
F | T | T
T | F | F
< | > | -->
---|---|-----------
T | F | F
F | T | T
F | F | T
T | T | T
< | > | -->
---|---|-----------
T | T | T
F | F | T
F | T | T
T | F | F
< | > | -->
---|---|-----------
T | T | T
F | T | T
F | F | T
T | F | F
< | > | -->
---|---|-----------
T | F | F
F | F | T
F | T | T
T | T | T
< | > | -->
---|---|-----------
F | F | T
T | T | T
T | F | F
F | T | T
< | > | -->
---|---|-----------
F | T | T
T | F | F
T | T | T
F | F | T
< | > | -->
---|---|-----------
F | T | T
T | T | T
T | F | F
F | F | T
< | > | -->
---|---|-----------
F | F | T
T | F | F
T | T | T
F | T | T
I have not worked out the details of the four-color logic involved because I needed to fix a context for
Zermelo-Fraenkel set theory first. Now I can start applying a topological analysis for threshold logic.
I doubt very much that any of this makes sense to you. It certainly doesn't to me based on any mathematics I learned
in school. But, it might explain why my posts are seemingly so incoherent. I am consciously aware of this context
management and am trying somewhat furiously at getting myself out of the situation by "navigating" carefully--whatever
that means.
Curiously, it may, in fact, be the case that my reasoning is extremely coherent.
:-)
mitch
Bill Taylor
indulging in further surrealism, neither do I...
Mike Oliver <oli...@math.ucla.edu> writes:
|> Well, write yourself into a bit stranger story.
I will! Thank you. But, knowing how little you like to draw sharp lines
between one thing and another, I expect you won't mind if I nudge
the scenario this way or that a little...
|> It's a formal derivation of tractable length -- let's say,
|> 10 pages or so of ordinary print. Every step has been checked line-by-line,
Let me reduce it to one page. And furthermore, let it be not merely
a formal derivation of an inconsistency, but an inconsistency of a specific
type, wherein all the lines are just numerical equations (or equivalent)
which can be individually tested.
My favourite sort of self-correction, when I start with an equation which
is clearly true, and finish up with one which is clearly false, is to track
down the error by bifurcation; (you'll have heard me lauding this method
before, I'm sure, in a different context! :) ) I try to impress this method
onto my students, but with little effect, alas. Dive into the middle,
and test the equation there - if it's OK go a quarter on, if it's not OK go
a quarter back. And so on till you find the mistake. It may of course
require making particular substitutions for general terms, but no problem.
It's very effective, quick, and a lot of fun!
Now I will do that with the page of alleged contradiction you gave me.
I eventually track down a mistake. But no - you won't allow me to, so
I must track it down to a line which is true, but it's following line
is false, and the implication is valid.
I'm sorry Mike, but I simply CANNOT CONCEIVE of that occurring!
It is just a nonsense. It is like asking me to swallow "= 2 + 2"
on one line and "= 5" on the next, and believing it!
It's just a rubbishment! Am I being unreasonable to insist on this?
I don't think so.
|> D) something else -- you fill it in.
So there it is. Over to you.
------------------------------------------------------------------------------
Bill Taylor W.Ta...@math.canterbury.ac.nz
------------------------------------------------------------------------------
There are only 10 kinds of people in the world -
Those who understand binary, and those who don't
------------------------------------------------------------------------------
Mike Oliver
>|> 10 pages or so of ordinary print. Every step has been checked line-by-line,
>
> Let me reduce it to one page. And furthermore, let it be not merely
> a formal derivation of an inconsistency, but an inconsistency of a specific
> type, wherein all the lines are just numerical equations (or equivalent)
> which can be individually tested.
I'd say this is quite a bit closer to the (B)-line, though I'm not
entirely sure whether I think it's crossed it. By "numerical equations"
you mean something like equalities between closed-form expressions
with +,* and decimal numbers, that will fit on a single line, right?
I expect (though I'm not sure) such an inconsistency in *those* could
be mechanically reduced to one in an equation in one- or two-digit numbers,
in not more than, I don't know, 100 or 1000 pages. So likely we could
reduce it to a contradiction without making an ontological commitment
to numbers we can't actually count to.
So no, this is not equivalent. I would like you to go back to
the original question, where the lines have quantifiers in them
and you can get from one to another with Delta_0 induction, and
it's allowed for one of the lines to be a formalization
of "10^10^10^10^10 exists".
Herman Jurjus
"Bill Taylor" <mat...@math.canterbury.ac.nz> wrote in message
news:anj0m0$6t8$1...@cantuc.canterbury.ac.nz...
> s...@sig.below (Barb Knox) writes:
[snip]
> |> > > A little philosophy is a dangerous thing...
> |> > Maybe. But it never knocked down any towers.
> |> Hmmmmm. Medieval Christianity, Islam, Marxism-Lenninism, ...
>
> HEY! Peeeeeeeeeeeeeeep! Thank you for playing. :)
>
> Those things are religions, not philosophy. Sure, there may be a little
> philosophy strirred into a religion, just like there may be some good
> health practices, but that doesn't make them the same. Religions have
> always been prone to mass murder and telling lies for god.
Hmm. How about this:
Christianity became murderous only after being infected with paganism and
Greek philosophy, iirc.
And let's face it: in history, non-believers have been much more agressive
towards believers than vice versa.
That, after a while, the remaining religions were all quite agressive, can
be blamed mostly on the military
powers of the ancient world. they phyisically extinguished the pacifict
variants...
(Take a look at the history of the Macabeans, for example.)
Quite OT, but i couldn't let this go by, sorry.
Cheers,
Herman Jurjus
John
> John wrote:
>
> > Interesting that Taylor doesn't cite *his own* account of Russell's
> > Paradox. In article news:<Bx2AL...@cantua.canterbury.ac.nz>
> > w...@math.canterbury.ac.nz (Bill Taylor) wrote:
> >
> > > The trouble with the Russell set, (or the universal set), is this.
> > > Attempting to define it by naive comprehension as R = { x | phi(x) },
> > > is objectionable; not (though) because phi may implicitly *refer to*
> > > itself, which is mere "harmless impredicativity", but because the
> > > **range of candidate members** includes R itself. This is the key
> > > to "vicious circle" impredicativity, it seems to me.
>
> > > To define a *set* (by comprehension), means to consider its
> > > possible members, so the set itself cannot possibly be considered
> > > as a prospective member, because it has not (yet) been defined.
> > > Doubtless this will sound just like "harmless impredicativity" to
> > > many, but I think it is a distinguishable special case, and the
> > > one that contains all the venom of the paradox.
>
> In at least one of my posts I recite:
>
> It makes no sense to say that an object is well
> defined iff and only if it is well defined.
Mitch, the problem with (1) is that it is *inconsistent*.
1) EyAx(xMy <-> ~(xMx))
And the inconsistency of (1) has nothing to do with sets, or
memberships, or vicious circles, or the Q-Moolah-Tive Hierarchy, or
anything of the sort. (1) is inconsistent because it proves a formula
and its negation. Correspondingly, the problem with (2-10) is that
each proves an instance of (1).
2) EyAx(x in y <-> Phi(x))
3) EyAx(x Taylors y <-> Phi(x))
4) EyAx(x Dullrichs y <-> Phi(x))
5) EyAx(x McCulloughs y <-> Phi(x))
6) EyAx(x Olivers y <-> Phi(x)))
7) EyAx(x Franzens y <-> Phi(x))
8) EyAx(x Greenes y <-> Phi(x))
9) EyAx(x Koskensiltas y <-> Phi(x))
10) EyAx(x Chapmans y <-> Phi(x))
What other differences there may be among (2-10) are of no
account to logic.
For Bill Taylor's diagnosis of RP--and my comments thereupon--
see news:<70f94e16.02100...@posting.google.com>.
--John
John
> John wrote:
>
> > Mitchell Smith <mit...@enteract.com> wrote in message news:<3D9DDBEA...@enteract.com>...
> > > Mitchell Smith wrote:
> > >
> > > > John wrote:
> > > >
> > > > Also, if you wish to respond to this, you may consider moving it to a different post. The material here could
> > > > lead to issues not germane to fundamental intuitions concerning the nature of sets (And, I say that with the
> > > > full realization that nobody is understanding my posts.).
> > >
> > > Dumb remark. Getting too tired again. Sorry. Any discussion of specific models of Zermelo-Fraenkel set theory
> > > are--by definition--germane to intuitions concerning the nature of sets.
> > >
> > > :-)
> > >
> > > mitch
> >
> > Take Bill Taylor and company with more than a grain of salt. And when
> > you're tired, get some sleep.
> >
> > --John
>
> Thanks.
>
> My exhaustion does not get resolved with sleep, unfortunately. Several years ago I was playing with syntactic
> forms--truth tables actually--and started to "organize" the forms. It turned into a game that does not stop.
Mitch,
I lack the background to understand any of this. Likewise, I have
no appreciation of the problems on which you say this bears.
So if you are looking for feedback, I'm sorry to say I won't be able to provide it.
--John
Mitchell Smith
John wrote:
> Mitch,
> I lack the background to understand any of this. Likewise, I have
> no appreciation of the problems on which you say this bears.
> So if you are looking for feedback, I'm sorry to say I won't be able to provide it.
>
Thank you for your candor.
After thinking about it last night, it occurs to me that it applies to the continuum question.
It is my understanding that our neural net expends the lion's share of its energy processing visual information. I am
beginning to suspect that there is a quaternary logic, presumably arises from subnets organized to act as linearly
separable threshold functions, which governs the kind of fixed contexts typical of formal systems.
It is not possible to derive this perspective by extracting forms from reasoned arguments because language users operate
with respect to the context fixed within their own sensory experience.
The question of consistency, however, forces language users to express self-referential forms, given that the model for
consistency they use is the self-consistency of their sensory experience.
Eventually, some poor bastard gets short-circuited.
:-)
mitch
Mitchell Smith
John wrote:
> Mitchell Smith <mit...@enteract.com> wrote in message news:<3D9E6F17...@enteract.com>...
> > In at least one of my posts I recite:
> >
> > It makes no sense to say that an object is well
> > defined iff and only if it is well defined.
>
> Mitch, the problem with (1) is that it is *inconsistent*.
>
> 1) EyAx(xMy <-> ~(xMx))
>
To be more precise, it is self-inconsistent.
That is a different issue from what Bill Taylor was *trying* to say
> > > Doubtless this will sound just like "harmless impredicativity" to
> > > many, but I think it is a distinguishable special case, and the
> > > one that contains all the venom of the paradox.
Look at the situation a little bit like statistical analysis of a null hypothesis.
There are *two* problems here. On the one hand, there is an irreconcilable self-inconsistent
logical form. On the other hand, whatever hyperlogical construct (x in x) represents, it is in
violation of a time-honored principle of mathematics called well definition.
Therefore...
If, as you did, an individual observes that the paradox arises from a self-inconsistent logical
form, that is a correct observation.
If, as I did, an individual observes that the inference taken from the paradox regarding the set
of all sets is fallible, that is a correct observation.
If, as others do, one concludes the paradox is attributed to the membership predicate, that is a
Type 1 error
If, as others do, one concludes that it is not possible to formulate a meaningful interpretation
of the universal quantifier in Zermelo-Fraenkel set theory as the class universe, that is a Type 2
error.
John
> Bill Taylor wrote:
> > |> I keep looking for a good opening to expound my thoughts in detail
> > |> on monotonic versus nonmonotonic knowledge;
> >
> > Mike - you keep teasing us with tidbits about this! I wish you would come clean.
> >
> > Now OC I don't want to rush you into anything incompletely cooked, but may I
> > at least reasonably request an advance preview? A trailer for the next film?
http://www.inos.com/users/pebbles/love_me_tender.ram
--John