ping Peter Percival

[mitch]

In another thread you asked about category theory
as a foundation for mathematics.

There is a very real sense by which Skolem's
criticisms of Zermelo explain why category
theory would be the next step from a "damaged
set theory". In trying to understand some of
Mr. Greene's views, I found the following SEP
article to be helpful,

http://plato.stanford.edu/entries/paradox-skolem/

Section 3 is the relevant section.

As explained there, Skolem seems to be calling
for an algebraic approach in his criticisms.
Then, set theory is "just another theory".

What this means is that universal algebra is
the foundation of mathematics.

To understand the significance of this for
category theory, first consider the Aristotelian
statement,

Man is an animal

One has the class of men being a part of the
class of animals.

Category theory is to algebras as term logic
is to classes. The nonsense about objects
and arrows refers to the groups defined by
group axioms and signatures, the rings defined
by ring axioms and signatures, and so on with
arrows referring to the various mappings that
preserve the structures.

So, treating set theory as "just another
theory" with respect to universal algebra
means that set theory is subsumed by category
theory in its treatment of algebras.

mitch

[graham...@gmail.com]

?- use_module(library(algebr)).
true.

?- 5+X = 7.
X = 2.

[Peter Percival]

mitch wrote:
>
> In another thread you asked about category theory
> as a foundation for mathematics.
>
> There is a very real sense by which Skolem's
> criticisms of Zermelo explain why category
> theory would be the next step from a "damaged
> set theory". In trying to understand some of
> Mr. Greene's views, I found the following SEP
> article to be helpful,
>
> http://plato.stanford.edu/entries/paradox-skolem/
>
> Section 3 is the relevant section.
>
> As explained there, Skolem seems to be calling
> for an algebraic approach in his criticisms.
> Then, set theory is "just another theory".
>
> What this means is that universal algebra is
> the foundation of mathematics.

I have Gratzer's /Universal algebra/ which I have only dipped into, but
it looks as if universal algebra rests a lot on set theory and logic
(especially model theory), so how it acts as a foundation (in the
nothing is logically prior to it sense) I don't know.

>
> To understand the significance of this for
> category theory, first consider the Aristotelian
> statement,
>
> Man is an animal
>
> One has the class of men being a part of the
> class of animals.
>
> Category theory is to algebras as term logic
> is to classes. The nonsense about objects
> and arrows refers to the groups defined by
> group axioms and signatures, the rings defined
> by ring axioms and signatures, and so on with
> arrows referring to the various mappings that
> preserve the structures.
>
> So, treating set theory as "just another
> theory" with respect to universal algebra
> means that set theory is subsumed by category
> theory in its treatment of algebras.

I have Gratzer's /Universal algbra


--
Do, as a concession to my poor wits, Lord Darlington, just explain
to me what you really mean.
I think I had better not, Duchess. Nowadays to be intelligible is
to be found out. -- Oscar Wilde, Lady Windermere's Fan

[Peter Percival]

Your point being...?

[mitch]

On 10/12/2016 09:25 AM, Peter Percival wrote:
> mitch wrote:
>>
>> In another thread you asked about category theory
>> as a foundation for mathematics.
>>
>> There is a very real sense by which Skolem's
>> criticisms of Zermelo explain why category
>> theory would be the next step from a "damaged
>> set theory". In trying to understand some of
>> Mr. Greene's views, I found the following SEP
>> article to be helpful,
>>
>> http://plato.stanford.edu/entries/paradox-skolem/
>>
>> Section 3 is the relevant section.
>>
>> As explained there, Skolem seems to be calling
>> for an algebraic approach in his criticisms.
>> Then, set theory is "just another theory".
>>
>> What this means is that universal algebra is
>> the foundation of mathematics.
>
> I have Gratzer's /Universal algebra/ which I have only dipped into, but
> it looks as if universal algebra rests a lot on set theory and logic
> (especially model theory), so how it acts as a foundation (in the
> nothing is logically prior to it sense) I don't know.

Nor do I.

In fact, I nearly fainted when Gratzer wrote that
universal algebra's best friend is lattice theory.

The axioms I study have been directed toward the
fact that the axioms of ZFC very nearly describe
a complete lattice. If it were a complete lattice,
the power set axiom would have a fixed point. This
would be Mr. Finlayson's "the universe is its own
power set" because the power set is not fixed
with respect to any other argument.

But, you need the universe to be an element of
itself so that there is a denotation under the
interpretation of the universal quantifier. And,
as I was so recently reminded by my misreading
of Rupert's post, this breaks reflection principles.

>>
>> To understand the significance of this for
>> category theory, first consider the Aristotelian
>> statement,
>>
>> Man is an animal
>>
>> One has the class of men being a part of the
>> class of animals.
>>
>> Category theory is to algebras as term logic
>> is to classes. The nonsense about objects
>> and arrows refers to the groups defined by
>> group axioms and signatures, the rings defined
>> by ring axioms and signatures, and so on with
>> arrows referring to the various mappings that
>> preserve the structures.
>>
>> So, treating set theory as "just another
>> theory" with respect to universal algebra
>> means that set theory is subsumed by category
>> theory in its treatment of algebras.
>
> I have Gratzer's /Universal algbra
>
>

So do I.

So, we both scratch our heads.

Hoping for some more guidance, I have picked
up "Sets for Mathematics" described by
Mr. Piitulainen on sci.math. The paragraph
titled "Foundation" reads,

< begin quote >

"A foundation makes explicit the essential general
features, ingredients, and operations of a science
as well as its origins and general laws of
development. The purpose of making these
explicit is to provide a guide to the learning,
use, and further development of the science.
A 'pure' foundation that forgets this purpose
and pursues a speculative 'foundation' for its
own sake is clearly a nonfoundation."

< end quote >

For me, at least, that is not specific as an
affirmative statement of what Lawvere and
Roseburgh mean by "foundation".

mitch

[mitch]

Why do you think mathematics is reducible to
Prolog, or some other high-level programming
language?

mitch

[Jack Campin]

> The axioms I study have been directed toward the
> fact that the axioms of ZFC very nearly describe
> a complete lattice. If it were a complete lattice,
> the power set axiom would have a fixed point. This
> would be Mr. Finlayson's "the universe is its own
> power set" because the power set is not fixed
> with respect to any other argument.

NF has a universal set, but your conclusion does not follow,
since the cardinality of the power set of V is strictly less
than that of V. Powerset doesn't have a fixed point in NF.

-----------------------------------------------------------------------------
e m a i l : j a c k @ c a m p i n . m e . u k
Jack Campin, 11 Third Street, Newtongrange, Midlothian EH22 4PU, Scotland
mobile 07800 739 557 <http://www.campin.me.uk> Twitter: JackCampin

[mitch]

On 10/15/2016 07:27 AM, Jack Campin wrote:
>> The axioms I study have been directed toward the fact that the
>> axioms of ZFC very nearly describe a complete lattice. If it were
>> a complete lattice, the power set axiom would have a fixed point.
>> This would be Mr. Finlayson's "the universe is its own power set"
>> because the power set is not fixed with respect to any other
>> argument.
>
> NF has a universal set, but your conclusion does not follow, since
> the cardinality of the power set of V is strictly less than that of
> V. Powerset doesn't have a fixed point in NF.
>

The axiom of choice fails in New Foundations. Hence, one
distinguishes between Cantorian and non-Cantorian classes.
That may have something to do with the result you are
stating. It is a very different system from ZFC. In
particular, the Quinean notion of an individual is based
upon lambda calculus principles. And, what is not
representable in New Foundations is the class of
individuals.

For what this is worth, Quine refers to his own theory
as a "class theory". I am not even sure that one ought
to refer to it as a "set theory".

With what I know of New Foundations from Quine's book,
"Set Theory and Its Logic", I would choose it as a better
account for a "logical" foundation foundation of mathematics.
To the extent that ZFC supposedly arises from Cantor, I
view the relationship of topology (via real analysis) to
the theory as something that cannot be ignored. To the
extent that it has been interpreted and developed otherwise in a
piecemeal fashion makes it complicated. But, the move away from
topology begins with Cantor. Arithmetization had been the way in
which mathematics was developing at that time, and, Cantor
developed an arithmetical theory from what began in
analysis. But, everything deriving from Cantor is very
different from "extensions of concepts" that is more
commonly associated with the tradition originating from
Frege and Russell. Quine's New Foundations seems to
extend that tradition in an appropriate manner. Cantor
justified "completed infinities" by analogy with ideal
points in non-Euclidean geometries. These were to be
understood as different types on the basis of how they
were generated. As a different type, a completed
infinity could be treated as a "unit" with respect to
that type. As a "unit", a completed infinity could
be replicated as the paradigmatic arithmetical unit
of standard arithmetic could. There is no "falling
under a concept" in this part of Cantor's development.

One reason I have issues with the relationship between
"first-order logic" and "purport" is the ease with which
proper classes are denoted with singular terms. Were my
interests purely logical, it would be easy for me to
suddenly accept "two-sorted" logic. But, my interpretation
of Russell's paradox is that the class cannot be formed.
So, pretending that a non-existence result justifies
a two-sorted logic in which a universe is miraculously
well-formed is a bit more than I can accept.

It may be that there is some paper with some analysis
that could resolve my discontent on this matter. I
have tried hard to find it. But, the mathematical,
logical, and philosophical literature is extensive. In
the meantime, I work on the problem with the tools I
have on hand.

One may consider individuating the universe in the
sense of the identity of indiscernibles. The universe
is precisely that class which differs from all others
in some way. But, thanks to Mr. Greene (legitimate
thanks), I have received a profound lesson about the
identity of indiscernibles. And, that is why I specifically
began working on a logic replacing the warrant provided
by the identity of indiscernibles with a logic whose
warrant comes from the assumptions of a theory (rather
than what is supposed to be a "mere" syntactic tranformation).

So, let me try to explain what I meant by my statement.

If the axioms of ZFC were to describe a complete lattice,
then a function over the universe would have to have
a fixed point, or, there would be something wrong
somewhere. Given that the power set operation cannot
be otherwise fixed, it could only be fixed for the
universe.

Why is the domain described by the axioms not a complete
lattice? It is because there is no analogue to the
axiom of union for the intersection operation. If there
were, intersection over the empty set would have to map
to the universe. But, for that to happen, it must be
possible to denote the universe as an element of the
domain.

There are two very fundamental presuppositions in this
analysis. First, it interprets the axiom of union as
"arbitrary unions" where "arbitrary" refers to whatever
exists (in a model). The definition of a complete lattice
depends upon "arbitrary union" and "arbitrary intersection".
This is why I say that a similar axiom for intersections
would be needed. The intersection over arbitrary sets
needs to be supported. ZFC has no such axiom, and, as a
response from Mr. Magidin on recent sci.math thread has
shown, the result is that one must suddenly interpret
intersections with respect to unions rather than with
respect to the universe of discourse. Second, I am
effectively claiming a subtle relationship between "set",
"element", "universe of discourse" and "denotation".

Whether or not my analysis has merit, the idea that
any symbol I write has existential import seems
problematic to me. This is what I mean when I refer
to "purport". One of the differences between Zermelo
and Skolem is that Zermelo *interprets* the sign of
equality with respect to membership in a singleton.
Zermelo's domain is based upon denotations. So, the
interpretation of equality between denotations is
grounded by the existence of singletons. Whereas
Skolem's papers may have been more appealing to his
peers and more reflective of beliefs in that era,
they contain no great development of any alternative.

Both Mr. Percival (if I am interpreting his remarks
faithfully) and I have difficulty seeing how the
algebraic view does not depend upon *warranting*
from its relationship with sets. For me, I have
traced that problem to Skolem's criticism of
Zermelo.

So, let me return to the problem of a universe.

If one could formulate an axiom of intersection to
interpret the universe as a complete lattice, the
reliance on singletons would still be problematic.
One would suddenly be trying to form classes that
included the universe as an element with simple
axioms like pairing and union. All that is really
required is that the universe be an element of
itself so that its denotation can be interpreted
as an element of the domain. And, this is a property
that no other class should have. The usual axioms
can be modified for this one exception.

So, just as I explained to Mr. Greene about the
"reasonableness" of the transfinite dimension
result, where is the "reasonableness" criterion
for what I am describing here? As I have pointed
out to Mr. Di Egidio, there are remarks in the
Wikipedia link on the specialization preorder
that apply,

https://en.wikipedia.org/wiki/Specialization_(pre)order

There is the explanation that the usage is consistent
with "genus" and "species" from classical logic.

There is the explanation concerning generic points
in algebraic geometry and how a generic point is
one contained in every non-empty open set. This
is to be compared with the particular point topology
and the closed extension topology,

https://en.wikipedia.org/wiki/Particular_point_topology

https://en.wikipedia.org/wiki/Extension_topology#Closed_extension_topology

In the section on important properties discussing
sober spaces,

https://en.wikipedia.org/wiki/Specialization_(pre)order#Important_properties

one has the remark,

< begin quote >

"One may describe the second property by saying that
open sets are inaccessible by directed suprema."

< end quote >

In other words, whether you call it omega or a
Woodin cardinal, if it can be put into a singleton,
it cannot "get there".

That is what has to be represented. And, you
cannot represent it with "purport". So, what I am
trying to talk about cannot be achieved by simply
introducing a constant and saying it represents
the universe of discourse.

In fact, because I have just been reading "Sets
in Mathematics" there is another passage in the
link on specialization preorder which applies to
foundations. Lawvere and Rosebrugh present the
"left adjoint" as a significant advance in the
history of foundations.

What all of this results from (in my case) is
that I wrote my axioms to have the negation of
the identity relation coincide with topological
inseparability,

https://en.wikipedia.org/wiki/Specialization_(pre)order#Important_properties

The standard account of identity does not include
the principle of identity of indiscernibles because
of historical arguments and debates. But, look at
axiom 2 in the link,

https://en.wikipedia.org/wiki/Metric_space#Definition

When I started my investigations there was no
internet and no wikipedia to interpret a metric
space axiom as the identity of indiscernibles.
Given this, that fact that there is now an internet
with a wikipedia page stating the obvious should
explain why people studying categories or homotopy
types are trying to claim a different foundation.

Compared to category theory, HOTT, MorTT (morphism
type theory), what I am doing is far simpler and
more integrated into foundational studies generally.
But, what I cannot defend against is the philosophical
problem that "words are meaningless" (except, of course
for those of the person who makes such an utterance).

So, what I know is that a function over a complete
lattice has a fixed point. If one takes that function
to be the power set operation and if one takes that
complete lattice to be the set universe, that fixed
point has to be the universe itself. Can this be
axiomatized without generating contradictions?

I don't know. But, I won't know if I do not try.

mitch

[Ross A. Finlayson]

That's quite remarkable, extended, and of
interest, to help illustrate that there's a
usual concern that a theory of everything
without a universe is at somewhat a loss in
explanatory power, while the theory with the
universe as fixed point yet edging to itself
rather than all the points toward it that that
demands a coordinated relaxation of the finite
principles and generators of the objects that
induction carries yet deduction results.


One notion that I've found helpful is that the
ur-element of theory, and this is really rather
general, is as well the empty or prototype
property-less element as it is the full or
prototype property-ful element. The true
foundations as about this primary object are
those results or inferences as about whether
it's either or both, then that it doesn't matter
the indiscernability which it is, because either
holds and is suited for inference.

This is very much then a conflation of sorts
of the "Nothing" and "Being" as of usual rather
deep (yet technical and grounded) philosophical
concerns as of the noumenon or Ding-an-Sich or
monad as of Kant or Leibniz, or some "Janus'
introspection", that Janus does have two faces
yet sees well from either, that the mirror yet
wouldn't reveal which one it was.

Then, about properties of the continuum, and,
individuation of elements, for analysis (and
real analysis or continuum analysis), there is
a didactic approach neatly enough as Dedekind
or Cauchy then in terms of Galois of the complete
ordered field so as to define an element as
propertied in the trichotomy of elements of the
reals, as a continuum. There is also a frequentist
approach, as to signal analysis, for the measurement
and observation effect, that our measurements of
real-valued variables is always as of an interval,
that the sampling frequency defines the precision
of the sample, that when resolved to a point, that
the sampling frequency is that of the rationals as
over the line (or dense in the lattice as of intersections
of connecting all the points in the lattice). As well,
there's a diagrammatic or geometric approach, besides
the algebraic and frequentist approach, that the
continuum is as of the segment and represents a scalar
magnitude or unsigned quantity as between zero and one.
This "field continuity", "signal continuity", and "line
continuity" have different definitions for models of
a continuum, and, the individuation(s) of their elements
so differ as well.

In metaphor of sorts to the "void" or "all" as either/or,
then generally some "foundation" (for inference) may be
as of sets or categories or otherwise objects defined by
their elements of membership or partitions and arrangements,
or there is a numerical foundation or as you note a
topological (or geometrical) foundation. These follow the
same from some least first principles of essentially some
"conservation of truth".

In some platonists' views: these foundations exist already
and are constant, complete, consistent, and concrete, then
for what is to be made of them overall (for essentially some
hierarchical reasoning) or made of them specifically (for
example, "ZF" or "NF", or finite combinatorics or a two-state
machine).

So, adherents of usual formalist axiomatizations might want
to revisit the notion that "naive" foundations are the more
sublime, and that restriction of comprehension or proof by
contradiction can be very useful tools, but there re the
consequences of Goedel, and that whatever true results there
are follow some facts of these "free foundations".


Then, Mitch, or as you'd be addressed, your picking a top-
down or another bottom-up approach has already these well-
known consequences of Goedel. So, for a universe that's
bigger than itself, _and a theory that's stronger than itself_
(and very resistant to meddling), why not have both and
reflect that they're either, thus one and both?

[graham...@gmail.com]

Well that wasn't the point. The point is logic came
across a stumbling block and became entrenched in Godel
Proof as a cyclic argument to move no further.

What formula is not transformed into a predicate?
What predicate A has no general form A->B ?
What logical reduction is made outside the rule of modus ponens?

Gödel's Proof is a 2 valued logic with self reference so
paradoxes will be evident. So you have all proved nothing!

Every paradox can be rewritten in common sense form where
the self reference is evident.

"Which formula asserts there is a proof of itself being wrong?"
"Which number set contains the numerical set indexes that are not self members?"
"Which fixed size turing machine outputs the maximum output size of all sized turing machines?"

the fundamental function is a matching predicate

f(A,B,c) == f(E,d,c)

A==E
B=d
c=c

MATCH or NO-MATCH

UNIFY( f(A,B,c) , f(E,d,c) )

or UNIFY (f(A,B,c) , { DATABASE OF PREDICATES } )

is the only function needed

[Ross A. Finlayson]

It's like: whether you're
an optimist or a pessimist,
the glass is half-empty and
half-full.

[mitch]

That symmetry is there.

Aristotle's term logic semantics had been based on
a part-whole relationship -- hence an order relation.
It is semantically prioritized with "individuals" as
"primary substance". Classes are "secondary substance".
When describing his own approach to logic, Leibniz
specifically says that he inverts the ordering of
classical logic.

This is why what I am trying now has a mereological
order and a predicative order. One is top-down and
the other is botttom-up.

mtich

[Ross A. Finlayson]

Euh, set theory, with classes, is set theory with classes.

When you say mereological then that's of boundaries,
as of the partitions instead of the collections. A
part theory is a reflection about a symmetry of a set
theory, that starts with a whole, for its parts, instead
of a hole, as a part (or here a thing that happens to
be a constant and is usually labelled and equated with
the empty set). Then, for the void and universal, that's
like the point and total, set theory and part theory.
That's just to illustrate it's this same reflection about
symmetry of equi-interpretability and transfer (or anti-
transfer). The resulting proofs are the same in their
results or as they purport, while their structure is
as of totally different compositions (and as opposites
that the one is other in the "mirror universe" of sorts,
or that the one is the "context" of the other).

Then having mereological operations, as they are, and
predicative operations, as they are, seems to introduce
primary deductive for inductive elements, of one for
the other and each for both (all quite constructivistically).

[mitch]

If I were to manage this, how would you call
a theory in which every element is an element?
That is, what if the theory admits only the
universe in addition to the sets of a cumulative
hierarchy? If the universe is an element of
itself, then it would seem to be only a set
theory.

But, I worry more about trying to formulate
it than I do about what to call it.

>
> When you say mereological then that's of boundaries,
> as of the partitions instead of the collections.

When I say mereological I also refer to Tarski's
paper on the biconditional as a primitive
connective and Lesniewski's analysis motivated
by that paper. This, in turn, goes right back
to Leibniz' identity of indiscernibles.

An order had two directions. Semantics arising
from individuals is bottom-to-top. Pedagogy
invoking universals is top-to-bottom. The
top-to-bottom direction is the one corresponding
to convergent filters. This is the identity of
indiscernibles.

The received paradigm in logic, however, has rejected
the identity of indiscernibles. So, the subset
relation must be predicative -- arising from
individuals as the extensional semantics with which
it is associated dictates.

In principle, the two orders may not be
distinguishable. But, the fact that non-structural
priorities generate two distinct asymmetries means
that they need to be treated separately.

In modern contexts, one is differentiating
between a maximal proper filter and a maximal
proper ideal. These differ only with respect
to the extremes of the order. The filter
has the top as an element. The ideal has the
bottom as an element.

> A
> part theory is a reflection about a symmetry of a set
> theory, that starts with a whole, for its parts, instead
> of a hole, as a part (or here a thing that happens to
> be a constant and is usually labelled and equated with
> the empty set). Then, for the void and universal, that's
> like the point and total, set theory and part theory.
> That's just to illustrate it's this same reflection about
> symmetry of equi-interpretability and transfer (or anti-
> transfer). The resulting proofs are the same in their
> results or as they purport, while their structure is
> as of totally different compositions (and as opposites
> that the one is other in the "mirror universe" of sorts,
> or that the one is the "context" of the other).
>
> Then having mereological operations, as they are, and
> predicative operations, as they are, seems to introduce
> primary deductive for inductive elements, of one for
> the other and each for both (all quite constructivistically).

A constructive view I never intended to have.

mitch

[Ross A. Finlayson]

When you say "rejected the identity of indiscernibles"
I think you mean that there are non-constructible
elements, not impredicative ones (eg that there are
uncountably many thus not a name for each that they
are discernible and as from each other, yet still
that each must exist so as to satisfy filling the
language of the structures and establishing identity
and non-identity).

Then, there are various concerns like "V = L", that
the universe is the constructible universe, where
the received paradigm (or today's dogmatic outlook)
does have a place for discernibles as the individuals
of a continuum, and, does have (another) place for
indiscernibles that yet maintain their identity.

That it might seem a consequence of inductive and
deductive reasoning about the bounds and extremes
of quantification usual structural features of
the extra-ordinary, it's still that those don't
prohibit the fixed point or as means of exhaustion,
where, this is known since antiquity as the paradoxes
of Zeno of Elea and there are usual considerations
as to their resolution (that there is the partitioning
and partitioned, to individua, or collecting and
collected, to continua).

Having "just the universe" as an extra-ordinary
element appended to an (otherwise) ordinary (or
well-founded with infinity) theory, seems instead
a role for a dedicated universal quantifier, which
most languages of the logic support while there is
not always explored so much the consequence of the
mere existence of the universal quantifier in an
ordinary theory. Then, this is often reflected upon
that this quantifier is "naive" and the theory is
really just a "schema" for bounded fragments of the
(soi-disant) theory.

Then I'd totally encourage you to have an extra-
ordinary theory for the application of logic as
from purely logical principles, I think that there
is such a thing (and as built from all the schemas
of fragments), this from usual least first principles
(none). I think there is this for you and for
anyone, and that it's compatible with the
conscientious constructivist, and formalists 24/7.

There are many applications yet to be made from
geometry and from the foundations, from the direct
properties of the continuum and as of the space of
numbers, that are simply enough beyond the foundations
of the day (for an improved returned paradigm).

[mitch]

The standard account of identity:

http://plato.stanford.edu/entries/identity-relative/#1

If you find the identity of indiscernibles hiding
there, let me know.

> I think you mean that there are non-constructible
> elements, not impredicative ones (eg that there are
> uncountably many thus not a name for each that they
> are discernible and as from each other, yet still
> that each must exist so as to satisfy filling the
> language of the structures and establishing identity
> and non-identity).

As I understand matters, Hilbert's concern had
been that mathematics should not contain any
inherent contradiction so that sciences depending
upon mathematics would not be subject to the
disruption a mathematical contradiction might
entail.

Consistency is one means (maybe the only means)
of demonstrating non-contradictoriness. But,
the existence of forms whose existence depends upon
the prior existence of individuals is problematic
for the universals used in proofs, and elsewhere.
The same holds of functions and relations.

I do not care about any plurality of non-set
denotations outside of the cumulative hierarchy.
I only care that a denotation for the universe
is supported under the interpretation of the
universal quantifier.

>

> Then, there are various concerns like "V = L", that
> the universe is the constructible universe, where
> the received paradigm (or today's dogmatic outlook)
> does have a place for discernibles as the individuals
> of a continuum, and, does have (another) place for
> indiscernibles that yet maintain their identity.
>
> That it might seem a consequence of inductive and
> deductive reasoning about the bounds and extremes
> of quantification usual structural features of
> the extra-ordinary, it's still that those don't
> prohibit the fixed point or as means of exhaustion,
> where, this is known since antiquity as the paradoxes
> of Zeno of Elea and there are usual considerations
> as to their resolution (that there is the partitioning
> and partitioned, to individua, or collecting and
> collected, to continua).
>

Cantor's second principle of counting basically
reinvents any fixed point into a unit from which
succession may proceed. So something intended to
exhaust succession is inadequate.

And, all set-theoretic forcing based upon the
generic forcing theorem assumes partiality. So,
anything that can be a part is inadequate.

> Having "just the universe" as an extra-ordinary
> element appended to an (otherwise) ordinary (or
> well-founded with infinity) theory, seems instead
> a role for a dedicated universal quantifier, which

> most languages of the logic supportwhile there is

> not always explored so much the consequence of the
> mere existence of the universal quantifier in an
> ordinary theory. Then, this is often reflected upon
> that this quantifier is "naive" and the theory is
> really just a "schema" for bounded fragments of the
> (soi-disant) theory.
>

nice word

I never saw it before.

> Then I'd totally encourage you to have an extra-
> ordinary theory for the application of logic as
> from purely logical principles, I think that there
> is such a thing (and as built from all the schemas
> of fragments), this from usual least first principles
> (none). I think there is this for you and for
> anyone, and that it's compatible with the
> conscientious constructivist, and formalists 24/7.
>
> There are many applications yet to be made from
> geometry and from the foundations, from the direct
> properties of the continuum and as of the space of
> numbers, that are simply enough beyond the foundations
> of the day (for an improved returned paradigm).
>

Well, there are now many paradigms. I
just watched Professor Bauer's video brought
to the attention of sci.math by FredJeffries.
A future I will not see will probably get
pretty interesting if it is not subsumed by
mechanical automata.

mitch

[Ross A. Finlayson]

Andrej Bauer as representing the Princeton IAS
conference on HTT, you mention now he has a
video, I imagine it's on "youtube".

There's usually much less to be gained from the
video or spoken presentation (except as in richer
inflection for the fluent), though a personal
communication may convey more information for
the learner with the detection of usual cues of
attention, distraction, comprehension, or confusion.

It seems there's quite a bit more as from his "blog".


Re "identity", again there's some reason to variegate
between identity, equality, tautology (as of the fixed
point). Here my usage of identity and equality are
reversed, say, from that, so instead I will reverse mine.

identity: x = f(y)
equality: x = x

This was from a today's computing where equality is as of
"logical equivalence" where identity was the value of the
reference. So, as above Leibniz has those reversed, more
or less. Then, tautology as I would put it is along the
lines of the fixed point or limit:

tautology: f(y) -> x

or that the l.h.s. and r.h.s "approach" each other.

tautology: f(y) -> <- f(x)

Then, this tautology is an equivalence, but doesn't
necessarily carry the indiscernability (equality under
all circumstances) of the lhs and rhs of the expression.

Then as above this is due constructibility or "naming"
concerns of objects, where in a set theory objects
are defined by their elements (as their structure and
also their name). Then, identity and tautology may
so build in defining equivalence classes of sets,
and they are quite discernible the lhs from the rhs
when those are different models on either side of
the equator, then quite indiscernible when they are
the same type (for example as Russell stratifies and
ramifies types, where he has hoisted type theory into
only a set theory).

The properties of symmetry (reflexitivity), indiscern-
ibility, then transitivity are variously attached to
these various forms of equivalence. In a usual enough
sense they are each properties that define what an
equivalence is, but under conditions, usually enough
at the ends, or in the course through the middle,
they variously do or don't hold. Clearly this
implies their generation or existence as of a
course-of-passage (ordinal) of a structural definition.

"Assume that a and b are individual constants having
the same interpretation in M1 and M2. Let E1 and E2
be the identity symbols of L1 and L2. It can happen
that E1(a,b) is true in M1 but E2(a,b) is false in M2.
We can then say, with Geach (1967; see §4) and others,
that the self-same objects indiscernible according to
one theory may be discernible according to another."

This again is already part of the consideration of
set theory as foundations that the incompleteness
of Goedel has there may be "true" features _about
the objects in the domain of discourse or the
domain of discourse itself already_.

As sets, a = b implies that a and b, defined by their
elements, are identical (one is an identity for
the other as an expression), equal (they are the
same value), and tautologous (they are the same member).

Here this code use a bit more formalism to codify
these properties in their establishment in usual
various normal forms.

The SEP article then continues into "The Paradox
of Change" (eg, the only constant is change, that
which is at rest stays at rest, probably versions
of the Sorites for the differential or the delta,
or the update problem as from the digital.)

[Ross A. Finlayson]

This then gets into quantifier differentiation
and why and where there is some reasonable,
meaningful, tractable difference between and
among:

for-any
for-every
for-each
for-all

with the goal of maintaining in simple, terse
terms these usual, systematic conventions (and
maintenance and obliterations thereof).

[mitch]

On 10/15/2016 07:27 AM, Jack Campin wrote:

>> The axioms I study have been directed toward the
>> fact that the axioms of ZFC very nearly describe
>> a complete lattice. If it were a complete lattice,
>> the power set axiom would have a fixed point. This
>> would be Mr. Finlayson's "the universe is its own
>> power set" because the power set is not fixed
>> with respect to any other argument.
>
> NF has a universal set, but your conclusion does not follow,
> since the cardinality of the power set of V is strictly less
> than that of V. Powerset doesn't have a fixed point in NF.
>

I have found an online paper with
excerpts describing exactly what I
a trying to represent.

In Section 2.3 of the paper,

http://www.filozof.uni.lodz.pl/bulletin/pdf/40_34_3.pdf

you will find a discussion of a
closure operation on a set without
complementation. The transitive
closures of ZFC satisfy this operation.

In section 2.5 you will see a discussion
of "well-connected" closure axioms. In
the section with the alternate axiom W1
you will see that it discusses intersection
over the empty family. That intersection
is the top (1) and is not the bottom (0).

What exists in a power set must exist
in the universe. For the power set of
a set formed with respect to the cumulative
hierarchy, power sets are systems of
Boolean complements because each of the
two complements exists in the cumulative
hierarchy.

To say that the universe is its own
power set is to say that its elements
are the sets.

In natural language the statement is
trivial. But, since it requires describing
a universe that is an element of itself,
formalizing it is no easy matter. That
requirement is introduced by intersection
over an empty family.

mitch

[mitch]

On 10/12/2016 09:25 AM, Peter Percival wrote:
> mitch wrote:
>>

< snip >

>>
>> So, treating set theory as "just another
>> theory" with respect to universal algebra
>> means that set theory is subsumed by category
>> theory in its treatment of algebras.
>
> I have Gratzer's /Universal algbra
>

While I have been looking for my set
theory books, I have been reading
"Sets for Mathematics" by Lawvere and
Rosebrugh. I could easily go through
a number of definitions intended to
re-interpret basic ideas. But, I have
only read the first chapter or two.

One thing that I have already identified
is that the program assumes the identity
of indiscernibles. This, of course, is
*not* part of the received paradigm. But
it is consistent with remarks I have made
elsewhere concerning the role of topology
in a standard mathematics curriculum.

Should you be interested, I would take
some time to present and discuss some
of these definitions.

But, in this response, I thought I would
share something from reading ahead when I
noticed a heading about Cantor:

< begin quote >

"Cantor's method for proving this theorem
is often called the 'diagonal argument'
even though the diagonal map delta_x is only
one of two equally necessary pillars on which
the argument stands, the second being a
fixed-point free self-map tau (such as logical
negation in the case of [the object] 2). This
diagonal argument has been traced (by philosophers)
back to ancient philosophers who used something
like it to mystify people with the Liar's
paradox. Cantor, however, used his method to
prove positive results, namely inequalities
between cardinalities. The philosopher Bertrand
Russell, who was familiar with Cantor's theorem,
applied it to demonstrate the inconsistency of
a system of logic proposed by the philosopher
Frege; since then philosophers have referred to
Cantor's theorem as Russell's paradox and have
even used their relapse into the ancient paradox
habit as a reason for their otherwise unfounded
rumor that Cantor's set theory might be
inconsistent. (Combating this rumor became one
of the main preoccupations of the developers of
axiomatized set theories of Zermelo, Fraenkel,
von Neumann, and Bernays. This preoccupation
assumed such an importance that the use of such
axiom systems for *clarifying* the role of
abstract sets as a *guide* to mathematical
subjects such as geometry, analysis, combinatorial
topology, etc., fell into neglect for many
years.)"

< end quote >

They do cite Suppes, "Axiomatic Set Theory" as
a source for some of this history. I no longer
have that book. So, I cannot trace the reference.

I want to make a joke here. I am afraid the
irony would be lost and I would be viewed as
saying one more insulting thing about logicians
and philosophers.

In any case, this sounds to me like they are
simply assuming consistency.

mitch