What are sets? again

[Zuhair]

The following is an account about what sets are, first I'll write the
exposition of this base theory in brief, then I'll discuss some
related issues.

Language: FOL + P, Rp

P stands for "is part of"

Rp stands for "represents"

Axioms: Identity theory axioms +

I. Part-hood: P partially orders the universe.

ll. Supplementation: x P y & ~ y P x -> Exist z. z P y & ~ x P z.

Def.) atom(x) <-> for all y. y P x -> x P y

Def.) x atom of y <-> atom(x) & x P y.

Def.) c is a collection of atoms iff for all y. y P c -> Exist z. z
atom of y.

Def.) c is atomless <-> ~ Exist x. x atom of c

lll. Representation: x Rp c & y Rp d -> (x=y<->c=d)

lV. Representatives: x Rp c -> atom(x)

V. Null: Exist! x. (Exist c. x Rp c & c is atomless).

A Set is an atom that uniquely represents a collection of atoms or
absence of atoms.

Def.) Set(x) <-> Exist c. (c is a collection of atoms or c is
atomless) & x Rp c & atom(x)

Here in this theory because of lV there is no need to mention atom(x)
in the above definition.

Set membership is being an atom of a collection of atoms that is
uniquely represented by an atom.

Def.) x e y iff Exist c. c is a collection of atoms & y Rp c & x atom
of c & atom(y)

Here in this theory because of lV there is no need to mention atom(y)
in the above definition.

Vl. Composition: if phi is a formula in which y is free but x not,
then
[Exist y. atom(y) & phi] -> [Exist x. x is a collection of atoms &
(for all y. y atom of x <-> atom(y) & phi)] is an axiom.

Vll. Pairing: for all atoms c,d Exist x for all y. y e x <-> y=c or
y=d
/

This theory can interpret second order arithmetic. And I like to think
of it as a base theory on top of which any stronger set theory can
have its axioms added to it relativized to sets and with set
membership defined as above, so for example one can add all ZFC axioms
in this manner, and the result would be a theory that defines a model
of ZFC, and thus proves the consistency of ZFC. Anyhow this would only
be a representation of those theories in terms of different
primitives, and it is justified if one think of those primitives as a
more natural than membership, or if one think that it is useful to
explicate the later. Moreover this method makes one see the Whole
Ontology involved with set\class theories, thus the bigger picture
revealed! This is not usually seen with set theories or even class
theories as usually presented, here one can see the interplay between
sets and classes (collections of atoms), and also one can easily add
Ur-elements to this theory and still be able to discriminate it from
the empty set at the same time, a simple approach is to stipulate the
existence of atoms that do not represent any object. It is also very
easy to explicate non well founded scenarios here in almost flawless
manner. Even gross violation of Extensionality can be easily
contemplated here. So most of different contexts involved with various
maneuvering with set\class theories can be easily
paralleled here and understood in almost naive manner.

In simple words the above approach speaks about sets as being atomic
representatives of collections (or absence) of atoms, the advantage is
clearly of obtaining a hierarchy of objects. Of course an atom here
refers to indivisible objects with respect to relation P here, and
this is just a descriptive atom-hood that depends on discourse of this
theory, it doesn't mean true atoms that physically have no parts, it
only means that in the discourse of this theory there
is no description of proper parts of them, so for example one can add
new primitive to this theory like for example the primitive "physical"
and stipulate that any physical object is an atom, so a city for
example would be an atom, it means it is descriptively an atom as far
as the discourse of this theory is concerned, so atom-hood is a
descriptive modality here. From this one can understand that a set is
a way to look at a collection of atoms from atomic perspective, so the
set is the atomic representative of that collection, i.e. it is what
one perceives when handling a collection of atoms as one descriptive
\discursive whole, this one descriptive\discursive whole is actually
the atom that uniquely represents that collection of atoms, and the
current methodology is meant to capture this concept.

Now from all of that it is clear that Set and Set membership are not
pure mathematical concepts, they are actually reflecting a
hierarchical interplay of the singular and the plural, which is at a
more basic level than mathematics, it is down at the level of Logic
actually, so it can be viewed as a powerful form of logic, even the
added axioms to the base theory above like those of ZFC are really
more general than being mathematical and even when mathematical
concepts are interpreted in it still the interpretation is not
completely faithful to those concepts. However this powerful logical
background does provide the necessary Ontology required for
mathematical objects to be secured and for
their rules to be checked for consistency.

But what constitutes mathematics? Which concepts if interpreted in the
above powerful kind of logic would be considered as mathematical? This
proves to be a very difficult question. I'm tending to think that
mathematics is nothing but "Discourse about abstract structure", where
abstract structure is a kind of free standing structural universal.
Anyhow I'm not sure of the later. I don't think anybody really
succeeded with carrying along such concepts.

Zuhair

[William Elliot]

On Fri, 30 Nov 2012, Zuhair wrote:

> The following is an account about what sets are,
>

> Language: FOL + P, Rp
> P stands for "is part of"
>

Does P represent "subset of" or "member of"?

> Rp stands for "represents"
>
Give an intuitive example or two how you interpreted "represents".

[Zuhair]

On Dec 1, 1:41 pm, William Elliot <ma...@panix.com> wrote:
> On Fri, 30 Nov 2012, Zuhair wrote:
> > The following is an account about what sets are,
>
> > Language: FOL + P, Rp
> > P stands for "is part of"
>
> Does P represent "subset of" or "member of"?
>

Neither.

P represents "is part of"
review mereology to understand that relation informally.

> > Rp stands for "represents"
>
> Give an intuitive example or two how you interpreted "represents".
>

Informally representation by representatives is what is meant here.
Like for example: an attorny representing the defendant, or in an
Ambassador representing his country, etc... A representative might be
among the collection it represents like for example in fathers
representing their families or it might not be like the attorny
example above.

Axioms of this theory further characterise both Part-hood relation and
Represention relation.

Zuhair

[William Elliot]

On Sat, 1 Dec 2012, Zuhair wrote:
> On Dec 1, 1:41 pm, William Elliot <ma...@panix.com> wrote:
> > On Fri, 30 Nov 2012, Zuhair wrote:

> > > The following is an account about what sets are,
> >
> > > Language: FOL + P, Rp
> > > P stands for "is part of"
> >
> > Does P represent "subset of" or "member of"?
> Neither.
>
> P represents "is part of"
> review mereology to understand that relation informally.
>

What is simple jargon, a brief intuitive description of "is a part of".

[Graham Cooper]

On Dec 2, 5:44 pm, William Elliot <ma...@panix.com> wrote:
> On Sat, 1 Dec 2012, Zuhair wrote:
> > On Dec 1, 1:41 pm, William Elliot <ma...@panix.com> wrote:
> > > On Fri, 30 Nov 2012, Zuhair wrote:
> > > > The following is an account about what sets are,
>
> > > > Language: FOL + P, Rp
> > > > P stands for "is part of"
>
> > > Does P represent "subset of" or "member of"?
> > Neither.
>
> > P represents "is part of"
> > review mereology to understand that relation informally.
>
> What is simple jargon, a brief intuitive description of "is a part of".
>
>

http://en.wikipedia.org/wiki/Mereology

the part-whole relation orders its universe, meaning that everything
is a part of itself (reflexivity), that a part of a part of a whole is
itself a part of that whole (transitivity), and that two distinct
entities cannot each be a part of the other (antisymmetry).

[Zuhair]

On Dec 2, 10:44 am, William Elliot <ma...@panix.com> wrote:
> On Sat, 1 Dec 2012, Zuhair wrote:
> > On Dec 1, 1:41 pm, William Elliot <ma...@panix.com> wrote:
> > > On Fri, 30 Nov 2012, Zuhair wrote:
> > > > The following is an account about what sets are,
>
> > > > Language: FOL + P, Rp
> > > > P stands for "is part of"
>
> > > Does P represent "subset of" or "member of"?
> > Neither.
>
> > P represents "is part of"
> > review mereology to understand that relation informally.
>
> What is simple jargon, a brief intuitive description of "is a part of".

Just read Varzi's article on Mereology:

http://plato.stanford.edu/entries/mereology/

The relation "is part of" is well understood philosophically speaking,
it has natural examples.
I think Varzi's account on it is nice and interesting really. You can
also read David Lewis account on it. The discipline of Mereology is
well established.

Zuhair

[Charlie-Boo]

All you need to do is to say that every FOL wff with a free variable
defines a set and then what are the base relations. "x is an element
of y" is not a relation, which you can say explicitly or leave it out
and so it is not implicitly. If you want to generate incompleteness
statements (e.g. Godel/Rosser/Smullyan or Turing Halting Problem) then
you need to specify it explicitly.

This would imply that subsets, unions, intersections, complement etc.
are all sets since they are represented as FOL wffs.

Defining "class" is folly. They don't exist and they don't solve
anything. It is just "kicking the can down the road." Your next
problem is whether or not the class of classes that don't contain
themselves contains itslef, and that is talking about something that
is totally arbitrary and artificial in the first place!

In every axiomatization that I have developed (about 6) whenever there
is incompleteness (2 or 3) it is axiomatized by a single axiom. This
resolves paradoxes and even lets you generate them. In one contrived
axiomatization every theorem is an incompleteness theorem (include the
above.)

C-B

[William Elliot]

On Sun, 2 Dec 2012, Zuhair wrote:
> On Dec 2, 10:44 am, William Elliot <ma...@panix.com> wrote:
> > On Sat, 1 Dec 2012, Zuhair wrote:

> > > > > The following is an account about what sets are,
> >
> > > > > Language: FOL + P, Rp
> > > > > P stands for "is part of"
> >
> > > > Does P represent "subset of" or "member of"?
> > > Neither.
> >
> > > P represents "is part of"
> > > review mereology to understand that relation informally.
> > What is simple jargon, a brief intuitive description of "is a part of".

> Just read Varzi's article on Mereology:
> http://plato.stanford.edu/entries/mereology/

It's long winded as philosophy usually is.
Basically, "is a part of" is a (partial) order.
"Subset" is the better interpretation that "is member of".

So I'll take it as "subset" unless you give a useful
interpretation within 300 words or less.

> The relation "is part of" is well understood philosophically speaking,
> it has natural examples.

For example?

> I think Varzi's account on it is nice and interesting really. You can
> also read David Lewis account on it. The discipline of Mereology is
> well established.

What's the point of mereology?

[Ross A. Finlayson]

Basically mereology is of the consideration of Brentano boundaries and
as comprehension in partitions or parts of wholes to complement
elements of sets, of the composition of things. It's a natural
complement to set theory.

What's the point of philosophy?

Regards,

Ross Finlayson

[William Elliot]

Oh of course; why wouldn't I think of that?

For example, the surface of Jupiter.

> What's the point of philosophy?

To befuddle erudite vagaries.

[William Elliot]

On Fri, 30 Nov 2012, Zuhair wrote:

> The following is an account about what sets are, first I'll write the
> exposition of this base theory in brief, then I'll discuss some
> related issues.
>
> Language: FOL + P, Rp
>
> P stands for "is part of"
>
> Rp stands for "represents"
>
> Axioms: Identity theory axioms +
>
> I. Part-hood: P partially orders the universe.
>
> ll. Supplementation: x P y & ~ y P x -> Exist z. z P y & ~ x P z.

x subset y, y not subset x -> some z subset y with x not subset z.
x proper subset y -> some z subset y with x not subset z
x proper subset y -> y\x subset y, x not subset y\x

Oh my, no empty set.

[Graham Cooper]

On Dec 3, 5:28 am, Charlie-Boo <shymath...@gmail.com> wrote:
>
> All you need to do is to say that every FOL wff with a free variable
> defines a set

But you need some extra syntax to specify which variable.


mod( mult( 2, X ) , 5 )

/\
||
\/

{ X | mod( mult( 2, X ) , 5 ) }


then you are back at Naive Set Theory.

Which is ok if you have some extra definitions to ensure some
consistency.

Herc

[fom]

Arguably, mereology as an investigation into foundational
mathematics is from Lesniewski

Lesniewski wrote several papers criticizing Russell's
Principia as basically being incoherent

He did his own investigations along the lines of logical
structure of sentences having existential import that
are different from both Frege and Russell

Subsequently, he characterized a notion of class that was
different from Russell's

At first, he tried to characterize his ideas in traditional
logical formats but ultimately began to pursue it
using formal syntax

This actually gets dense and I have not really
examined it. But, for example he begins a system
he calls protothetic with

A1. ((p <-> r) <-> (q <-> p)) <-> (r <-> q)

A2. (p <-> (q <-> r)) <-> ((p <-> q) <-> r)

and then lists 79 theorems about logical equivalence.

He then switches notation to quantify over propositional
variables

A1. ApAqAr(((p <-> r) <-> (q <-> p)) <-> (r <-> q))

A2. ApAqAr((p <-> (q <-> r)) <-> ((p <-> q) <-> r))

So that he can extend the system using variables ranging
over truth-functions

A3. AGAp(AF(G(p,p) <->
((Ar(F(r,r) <-> G(p,p))
<->
(Ar(F(r,r) <-> G(p <-> Aq(q),p))))
<-> Aq(G(q,p)))

He then lists 422 theorems in order to obtain the
three logical axioms of Lukasiewicz grounding
the usual theory of deduction based on implication
and negation. His primary stated goal at the
outset for extending the original system was to
obtain

ApAqAF((p <-> q) <-> (F(p) <-> F(q)))

and the equivalent was obtained at step 381

Next, he turns to his system of ontology
grounded upon the logical foundation of his
protothetic. His only axiom is

A0. AZAz((Z class of z) <-> (
(-(AY(-(Y class of Z)))
/\
AYAX(((Y class of Z) /\ (X class of Z)) -> (Y class of X)))
/\
AY((Y class of Z) -> (Y class of z))
))


In his exposition he comments on Russell's
paradox:

"... can be strengthened in ontology by
means of the easily proved sentence which
says that:

AZAz((Z class of z) -> (Z class of Z))


(I call this the 'ontological identity sentence';
it should be noticed that the yet stronger thesis

AZ(Z class of Z)

is not provable in ontology -- indeed, its
negation is provable.) In connection with this
sentence, I want to emphasize expressly that in
ontology there is always a very good possibility
of proving theses having a single component of
the type (Z class of Z) or (what is indifferently
the same in ontology) (z class of z). This does
not, however, lead to a contradiction via the
well-known schema of Principia Mathematica
because the definition directives of ontology
have been appropriately formulated so that
no thesis of the type

AZ((Z class of x) <-> -(Z class of Z))

can be obtained."


This assertion might best be viewed much like
the situation with general relativity. Philosophers
who have been looking at Lesniewski's systems
have not run into any contradictions such as
Russell's (at least, in so far as Peter Simons
has reported accurately)

The character of his predicate in ontology
allows him to formulate terminological
explanations about ontology within the language
of ontology.

The axiom of ontology, and the equivalent axioms
he discusses in his exposition derive from
his analysis of Russell's paradox.

The Lesniewskian notion of class is based upon
a part relation and this is the formal mereology
associated with his investigations:


A1:
If P is a part of object Q, then Q is not a
part of object P

A2:
If P is a part of object Q, and Q is a part of object R,
then P is a part of object R

D1:
P is an ingredient of an object Q when and only when,
P is the same object as Q or is a part of object Q

D2:
P is the class of objects p, when and only when the
following conditions are fulfilled:

a)
P is an object

b)
every p is an ingredient of object P

c)
for any Q, if Q is an ingredient of object P, then
some ingredient of object Q is an ingredient of
some p



These conditions formalize a statement Lesniewski
made in his analysis of Russell's paradox:

"I use the expressions "the set of all objects m"
and "the class of all objects m" to denote every
object P which fulfills the two following
conditions:

1) every m is an ingredient of the object P

2) if I is an ingredient of object P, then
some ingredient of object I is an ingredient
of some m"



Since ingredient is effectively the reflexive
subset relation in set theory by Zermelo's
1908 language, you can see why Zuhair chose the
language he did to describe atoms. In an
atomistic theory, I and m must at least share
some atom of P

I expressed this idea in a formal sense long
ago only to be flamed (singed by you and firebombed
by someone else)

What actually makes mereology work is something that
is associated with the constructible universe. It is
called almost universality. Of course, that is not
how Lesniewski referred to it:

"Lukasiewicz writes in his book as follows: 'we say
of objects belonging to a particular class, that
they are subordinated to that class'

"It most often happens that a class is not subordinated
to itself, as being a collection of elements, it
generally possesses different features from each of
its elements separately. A collection of men is not
a man, a collection of triangles is not a triangle,
etc. In some cases, it happens in fact to be otherwise.
Let us consider e.g., the conception of a 'full class',
i.e., a class to which belong, in general, some
individuals. For not all classes are full, some
being empty; e.g., the classes: "mountain of pure
gold', 'perpetual motion machine', 'square circle',
are empty, because there are no individuals which
belong to those classes. One can then distinguish
among them those classes to which belong some
individuals, and form the conception of a 'full
class'. Under this conception fall, as individuals,
whose classes, e.g., the class of men, the class
of triangles, the class of first even number (which
contains only one element, the number 2), etc.
A collection of all those classes constitutes
a new class, namely 'the class of full classes'.
So that the class of full classes is also a full
class and therefore is subordinated to itself."

In almost universal models of set theory, every subclass of
the universe is an element of the universe. Thus,
"is an element or is equal to" is the same as
"is a proper part or is equal to". So, the satisfaction
predicate can be reflexive containment... except for
one thing. The identity predicate in the theory can
not be based on extensionality. It must be based on
first-order object identity as described by Frege and
this cannot arise just because one invokes the
ontological position of a "theory of identity". The
reason that it must be based on object identity is that
reference to the universe can only be made if, as
Lesniewski has observed

P is an object

In an almost universal model, every proper part
of the universe is an element of a class of which
the universe is not an element. Thus every proper
part of the universe is distinguished from the
universe on the basis of object identity. Since
to be a 'full class' the universe can have no
other parts, it is unique and may be denoted
by a singular term.

I have a very strong suspicion that no one will
ever derive a Russellian paradox in Lesniewskian
mereology. This is especially true if one considers
George Greene's explanation that the paradox arises
from grammatical form. As we have seen, Lesniewski
specifically devised his mereology to circumvent the
grammatical forms he thought would be problematic.

[fom]

THIS IS REPOSTED TO THE THREAD
I HAD A CONFUSING NNTP CLIENT A
COUPLE OF DAYS AGO AND ACCIDENTLY
POSTED OUTSIDE OF THE THREAD

On Fri, 30 Nov 2012, Zuhair wrote:

> The following is an account about what sets are,
>
> Language: FOL + P, Rp
> P stands for "is part of"
>

> Rp stands for "represents"

>
> Axioms: Identity theory axioms +
>
> I. Part-hood: P partially orders the universe.

Typically, one cannot simply invoke a partial order

Thus P might be axiomatized as

Reflexiveness
Ax(xPx)

Transitivity
AxAyAz((xPy /\ yPz) -> xPz)


Anti-Symmetry
AxAy((xPy /\ yPx) -> x=y)

> On Sat, 01 Dec 2012, William Eliot asked:

>
>Does P represent "subset of" or "member of"?

His text indicates that he is attempting to apply
mereology to represent set theory. Thus, this
part relation is intended to coincide with
the (reflexive) subset relation.

On Fri, 30 Nov 2012, Zuhair wrote:

> ll. Supplementation: x P y & ~ y P x -> Exist z. z P y & ~ x P z.
>

This is not the typical notion of supplementation

Suppose x is such that zPx and -xPz
Then zPx and xPy imply zPy by transitivity
The proposed expression is satisfied but
supplementation is not established

Weak supplementation may be characterized using
an overlap relation and a disjointedness relation

AxAy(xOy <-> Ez(zPx /\ zPy))

AxAy(xDy <-> -xOy)

Then one forms the expression

AxAy((xPy /\ -yPx) -> Ez((zPy /\ -yPz) /\ zDx))

to express weak supplementation

On Fri, 30 Nov 2012, Zuhair wrote:

> Def.) atom(x) <-> for all y. y P x -> x P y
>

This is easier to understand by considering
the failure of the expression

Ey(yPx /\ -xPy)

which says that x has a proper part.

That is, using S (strict) for proper part,

AxAy(xSy <-> (xPy /\ -yPx))

a more typical definition of atom is
obtained from

Ax(atom(x) <-> -Ey(ySx))

> Def.) x atom of y <-> atom(x) & x P y.
>

Overloading works for certain programming languages,
but might be confusing in this context. What
about

AxAy(xMy <-> (atom(x) /\ xPy))
where M is read "is a monad of"

... anything but atom(x) and atom(x,y)




This is your pre-theoretic version of
classical set-theoretic extensionality

Typically, the right side is part of the axiom,

AxEy(atom(y) /\ yPx)

but your desire to introduce a null part is
not compatible with the ontological position
of mereologists. Thus, you formulate a
relation attaching atomism to each proper part
of the mereological universe independently

What you are doing here should be
compared to the restricted quantification
that underlies Goedel's constructible universe

Restricted quantifiers are expressed
by

E[uex]
A[uex]

So, the inductive step for describing satisfaction
of these quantifiers expands the syntax thusly,

M |= E[uex]psi(u,x,...)

is the same as

M |= Eu(uex /\ psi(u,x,...))

is the same as

(E[ueM])(uex /\ (M |= psi(u,x,...)))



These expressions are from Jech and the explanation
is as follows,

"M|=phi is obtained from phi by replacing
Ex and Ax by E[xeM] and A[xeM]. In particular,
if phi is quantifier free, (M|=phi) is the
same as phi"

Since Russell's distinction between apparent variables
(that is, bound variables) and real variables (that is,
unbound parameters) is not in play, this reference to
a quantifier free formula reflects an assumption in
Goedel's original paper,

"1. ...

2. Symbols a_1, ..., a_n denoting^(see footnote)
individual elements of M (referred to in the
sequel as 'the constants of phi')

3. ..."


footnote:

"It is assumed that for any element of M a symbol
denoting it can be introduced"


So, there are a great many presuppositions associated
with your construction if my comparison is
correct in its essentials.

The role played by restricted quantification
pertaining to the constructible universe
is given by the lemma,

If M is a transitive class and phi a
restricted formula, then for all
x_1, ..., x_n e M,

M|=phi(x_1, ..., x_n) iff phi(x_1, ..., x_n)


the corollary,

If M is a closed, transitive class, then M satisfies
restricted separation


and the theorem

Let M be a transitive, closed, almost universal
class. Then M is a model of ZF

On Fri, 30 Nov 2012, Zuhair wrote:

> Def.) c is a collection of atoms iff for all y. y P c -> Exist z. z
> atom of y.

Rewrite this as

Ax(collection(x) <-> Ay(yPx -> Ez(zMy)))



I once described this assertion as the membership
relation exhibiting an "object semantics" on the
left and a "collection semantics" on the right.

It was badly flamed.

It would have been flamed just as badly had I
used the phrases "object context" and
"collection context"

You have done a nice job of developing monadic
predicates to express this fact.

On Fri, 30 Nov 2012, Zuhair wrote:

> Def.) c is atomless <-> ~ Exist x. x atom of c
>

Rewrite this as

Ax(atomless(x) <-> -Ey(yMx))

Once again, you are trying to circumvent
the ontological position of mereology
to accommodate the null set

A typical hybrid mereological universe
might assume

Ex(atom(x) /\ ExAy(yPx -> Ez(zSy))

whereby "atomless" is associated with an infinitary
descending tree (by supplementation) of proper
parts (S - strict parts)

As I do not see that anything else in your system
precludes this possibility, you should probably
amend this to

Ax(atomless(x) <-> (-Ey(yMx) /\ -Ey(ySx)))

You may however be running into the ontological
problem of non-self-identicals

atom(x) "works" because the part relation is
reflexive

A classical mereology based on the extensionality
of parts would make

Ex(atomless(x))

unique as referring to the only object without
parts, but that ignores the ontological
position of mereology.

So, this may work for what you are trying to do.

On Fri, 30 Nov 2012, Zuhair wrote:

> lll. Representation: x Rp c & y Rp d -> (x=y<->c=d)
>
> lV. Representatives: x Rp c -> atom(x)
>

I was once flamed for not having read Zermelo's 1908 paper.

This paper is distinguished from later work on set theory
because denotation had still been significant to the historical
period in which the research was done.

Almost all modern set theory is presented as speaking of
objects and the explanation for the reflexive axiom in the
theory of identity is the self-identity of objects.

Let us call this view the objectual (Morris) or, more
technically, the ontological (Cocchiarella) position
that at least one author (Cocchiarella) associates
with Wittgenstein's Tractatus.

In 1908 Zermelo wrote:

"Set theory is concerned with a domain B
of individuals, which we shall call simply
objects and among which are the sets. If
two symbols, a and b, denote the same
object, we write a=b, otherwise -(a=b)."

So, Zermelo is considering the sign of equality
in terms of denotations. Consequently, the
reflexive axiom in the theory of identity
ought to be interpreted as asserting that
definite singular terms refer/represent
uniquely.

Let us call this view the metalinguistic (Morris) or,
more technically, the semantic (Cocchiarella) position
that at least one author (Cocchiarella, Morris)
associates with Frege's analyses.

Historically, Zermelo had been criticized
for not adequately characterizing his
notion of "definite" which later simply came
to mean expressed in formal language (as
suggested by Skolem among others).

But, Zermelo had been clear about "definite
identity",

"The question whether a=b or not is
always definite, since it is equivalent
to the question whether or not ae{b}"

(ae{b} is "a is an element of {b}" here)



In view of your explanation of representation,
it is reasonable to interpret singletons
as the atoms which act as representations.
This coincides with your notions of atoms
and how you are using them to formulate
representations.

Thus, the sequence

a, {a}, {{a}}, ...

is a sequence of Zermelo names rather
than Zermelo numbers.

The problem is thatthere is a
presupposition of a globally
consistent labelling. This is
needed because

ae{b}
be{a}

requires a criterion for a canonical
representation.

One can, from the ontological position, say
that labelling has nothing to do with it, but
that is not correct. The ontological position
effectively equates names with objects, and,
thereby, naming with individuation.

However one wishes to approach the question
of naming, the uniqueness demanded by Frege
and the semiotic constraints discussed by
Bolzano combine to demand that any
interpretation of naming presuppose a
canonical well-ordering of principal
names.

That this is presupposed by Goedel is
evidenced from the quote given above
that ends with

"... can be introduced."

rather than

"... is assumed."

If one wishes to assert this fact in
set theory without talking about
"labelling" one requires that the
axioms of set theory be strengthened
with an axiom asserting a global
axiom of choice. This is, in fact,
a property of the constructible universe.


Nice job on this idea.

On Fri, 30 Nov 2012, Zuhair wrote:

> V. Null: Exist! x. (Exist c. x Rp c & c is atomless).
>

Let's see...

Using N for your Rp

Ax(x=null() <-> Ay(atomless(y) /\ Az(atomless(z) -> y=z) /\ xNy))
Ey(atomless(y) /\ Az(atomless(z) -> y=z) /\ xNy)

as you are intending to introduce ur-elements as atoms

On Fri, 30 Nov 2012, Zuhair wrote:

> A Set is an atom that uniquely represents a collection of atoms or
> absence of atoms.
>
> Def.) Set(x) <-> Exist c. (c is a collection of atoms or c is
> atomless) & x Rp c

So, using xRpc you establish that a set has
an individuated context relative
to an atom.

But, actually this confuses me.

Using N for your Rp, I would think more in terms of

Ax(set(x) <-> ((collection(x) \/ atomless(x)) /\ Ey(yNx)))

Since one can form distinct expressions
purporting to refer to the same set, your
system must admit a plurality of atoms to
accommodate this fact. Treating the
representing atom as the canonical representation
means simply that its existence suffices
to ground the associated collection as
a set.

On Fri, 30 Nov 2012, Zuhair wrote:

> Def.) x e y iff Exist c. c is a collection of atoms & y Rp c & x atom
> of c
>

The same considerations as above apply here.

Using N for your Rp, I could see something along the lines of

AxAy(xey <-> (Ez(zNy) /\ collection(y) /\ xMy))

that would be interpretable from my perspective
(which is not necessarily correct).

On Fri, 30 Nov 2012, Zuhair wrote:

> Vl. Composition: if phi is a formula in which y is free but x not,
> then
> [Exist y. atom(y) & phi] -> [Exist x. x is a collection of atoms &
> (for all y. y atom of x <-> atom(y) & phi)] is an axiom.
>

Once again, two different expressions purporting
to refer to the same set seems problematic to
me.

On Fri, 30 Nov 2012, Zuhair wrote:

> Vll. Pairing: for all atoms c,d Exist x for all y. y e x <-> y=c or
> y=d
>

This should be unproblematic.

On Fri, 30 Nov 2012, Zuhair wrote:

> also one can easily add
> Ur-elements to this theory and still be able to discriminate it from
> the empty set at the same time, a simple approach is to stipulate the
> existence of atoms that do not represent any object.

So, taking your definition,

> Def.) x e y iff Exist c. c is a collection of atoms & y Rp c & x atom
> of c

a ur-atom, being an atom, is a collection of atoms
a ur-atom is also an atom of itself
but yRpc implies that y is an atom that represents

y, being an atom, is a collection of atoms
y is also an atom of itself
but, is y represented?

You have indicated features of representation which I
could understand and visualize in standard ZF, albeit
different from your intentions. But, you have not
included anything along the lines of

Ax(atom(x) -> Ey(yNx))

that would seem to be necessary

That is, one would expect

> Def.) Set(x) <-> Exist c. (c is a collection of atoms or c is
> atomless) & x Rp c & atom(x)

to be represented in the system

So, reverting to my visualization

c is the collection taken to be represented by a set
{c} is the atom representing c, so Set({c})
{{c}} is the atom representing {c}, so Set({{c}})
and so on


So, this

Ax(atom(x) -> Ey(yNx))

seems necessary

[fom]

On 12/2/2012 11:20 PM, William Elliot wrote:
> On Fri, 30 Nov 2012, Zuhair wrote:
>

<snip>

>> ll. Supplementation: x P y & ~ y P x -> Exist z. z P y & ~ x P z.
>
> x subset y, y not subset x -> some z subset y with x not subset z.
> x proper subset y -> some z subset y with x not subset z
> x proper subset y -> y\x subset y, x not subset y\x
>
> Oh my, no empty set.

You have made an incorrect step here.

In mereology there is no reason to take y\x as substantive.

Supplementation is supposed to enforce existence of a proper part of y
in y\x.

In this case, z could be a proper part of x. Then zPy and -xPz is
satisfied.

This is not a supplementation axiom in the classical sense.

As for no empty set, Zuhair may have seen this axiom in a formulation of
mereology where the axiom was intended to preclude existence of a null
part. This is a standard ontological position among those individuals
who investigate and reflects a position once taken by Frege in
criticizing the likes of Hausdorff and Cantor:

"... a forest without trees."

Moreover, Zuhair's construction is similar to Zermelo's 1908 paper on
set theory. Heijenoort's translates Zermelo's "Teil" -- that is,
subsets of nonvoid sets -- as "parts", and, the null set is introduced
separately.

This is precisely what Zuhair has attempted to do.

<snip>

[Zuhair]

On Dec 5, 5:06 am, fom <fomJ...@nyms.net> wrote:
> On 12/2/2012 11:20 PM, William Elliot wrote:
>
> > On Fri, 30 Nov 2012, Zuhair wrote:
>
> <snip>
>
> >> ll. Supplementation: x P y & ~ y P x -> Exist z. z P y & ~ x P z.
>
> > x subset y, y not subset x -> some z subset y with x not subset z.
> > x proper subset y -> some z subset y with x not subset z
> > x proper subset y -> y\x subset y, x not subset y\x
>
> > Oh my, no empty set.
>
> You have made an incorrect step here.
>
> In mereology there is no reason to take y\x as substantive.
>
> Supplementation is supposed to enforce existence of a proper part of y
> in y\x.
>
> In this case, z could be a proper part of x.  Then zPy and -xPz is
> satisfied.
>
> This is not a supplementation axiom in the classical sense.
>

I'm really sorry that I didn't have the chance to look at all of your
responses. I'd do once I have time.
Anyhow for now, it is sufficient to note that my theory does prove
Weak supplementation for collections of atoms that is if x is a proper
part of y and y is a collection of atoms then there exist a part of y
that do not overlap with x.

Zuhair

[Zuhair]

On Dec 5, 5:06 am, fom <fomJ...@nyms.net> wrote:

> On 12/2/2012 11:20 PM, William Elliot wrote:
>
> > On Fri, 30 Nov 2012, Zuhair wrote:
>
> <snip>
>
> >> ll. Supplementation: x P y & ~ y P x -> Exist z. z P y & ~ x P z.
>
> > x subset y, y not subset x -> some z subset y with x not subset z.
> > x proper subset y -> some z subset y with x not subset z
> > x proper subset y -> y\x subset y, x not subset y\x
>
> > Oh my, no empty set.
>
> You have made an incorrect step here.
>
> In mereology there is no reason to take y\x as substantive.
>
> Supplementation is supposed to enforce existence of a proper part of y
> in y\x.
>
> In this case, z could be a proper part of x.  Then zPy and -xPz is
> satisfied.
>
> This is not a supplementation axiom in the classical sense.
>

Correct. However in this theory weak supplementation is provable for
collections of atoms.

[Zuhair]

If the above system doesn't prove the following, then it must be
axiomatized.

x,z are collections of atoms -> [(for all y. y atom of x -> y atom of
z) -> x P z]

Zuhair

[fom]

On 12/4/2012 10:02 PM, Zuhair wrote:
> On Dec 5, 5:06 am, fom <fomJ...@nyms.net> wrote:
>> On 12/2/2012 11:20 PM, William Elliot wrote:
>>
>>> On Fri, 30 Nov 2012, Zuhair wrote:
>>
>> <snip>
>>
>>>> ll. Supplementation: x P y & ~ y P x -> Exist z. z P y & ~ x P z.
>>
>>> x subset y, y not subset x -> some z subset y with x not subset z.
>>> x proper subset y -> some z subset y with x not subset z
>>> x proper subset y -> y\x subset y, x not subset y\x
>>
>>> Oh my, no empty set.
>>
>> You have made an incorrect step here.
>>
>> In mereology there is no reason to take y\x as substantive.
>>
>> Supplementation is supposed to enforce existence of a proper part of y
>> in y\x.
>>
>> In this case, z could be a proper part of x. Then zPy and -xPz is
>> satisfied.
>>
>> This is not a supplementation axiom in the classical sense.
>>
>
> I'm really sorry that I didn't have the chance to look at all of your
> responses. I'd do once I have time.
> Anyhow for now, it is sufficient to note that my theory does prove
> Weak supplementation for collections of atoms that is if x is a proper
> part of y and y is a collection of atoms then there exist a part of y
> that do not overlap with x.
>
> Zuhair
>

Yes.

I can see that that should work with what you have done, although
I will not take the time to prove it for myself.

Then, of course, your null atom is simply a distinguished atom
in a theory that respects no empty class.

Don't worry to much about my responses. In part, I was rewriting
your sentences as part of an attempt to understand what you were
doing relative to my own meager knowledge.

Anyway, George will begin flaming me soon enough...

[Zuhair]

hmmm..., I see that I might have been wrong really. You seem to be
right.

What is needed is actually Weak supplementation, which is:

ll. Supplementation: x PP y -> Exist z. z P y & ~ z O x.

where z O x <-> Exist v. v P z & v P x

Zuhair

[Zuhair]

Language: FOL + P, Rp

P stands for "is part of"
Rp stands for "represents"

Axioms: Identity theory axioms +

I. Part-hood: P partially orders the universe.

ll. Supplementation: x P y & ~ y P x -> Exist z. z P y & ~ Exist v. v
P x & v P z.

Def.) atom(x) <-> for all y. y P x -> x P y
Def.) x atom of y <-> atom(x) & x P y.
Def.) c is a collection of atoms iff [for all y. y P c -> Exist z. z

atom of y].

Def.) c is atomless <-> ~ Exist x. x atom of c

lll. Representation: x Rp c & y Rp d -> (x=y<->c=d)
lV. Representatives: x Rp c -> atom(x)
V. Null: Exist! x. (Exist c. x Rp c & c is atomless)

A Set is an atom that uniquely represents a collection of atoms or
absence of atoms.

Def.) Set(x) <-> Exist c. (c is a collection of atoms or c is
atomless) & x Rp c & atom(x)

Here in this theory because of lV there is no need to mention atom(x)
in the above definition.

Set membership is being an atom of a collection of atoms that is
uniquely represented by an atom.

Def.) x e y iff Exist c. c is a collection of atoms & y Rp c & x atom
of c & atom(y)

Here in this theory because of lV there is no need to mention atom(y)
in the above definition.

Vl. Composition: if phi is a formula in which y is free but x not,
then [Exist y. atom(y) & phi] -> [Exist x. x is a collection of atoms
&
(for all y. y atom of x <-> atom(y) & phi)] is an axiom.

Vll. Pairing: for all atoms c,d Exist x for all y. y e x <-> y=c or
y=d

/

Zuhair

[Charlie-Boo]

On Dec 1, 8:46 am, Zuhair <zaljo...@gmail.com> wrote:
> On Dec 1, 1:41 pm, William Elliot <ma...@panix.com> wrote:> On Fri, 30 Nov 2012, Zuhair wrote:
> > > The following is an account about what sets are,
>
> > > Language: FOL + P, Rp
> > > P stands for "is part of"
>
> > Does P represent "subset of" or "member of"?
>
> Neither.
>
> P represents "is part of"
> review mereology to understand that relation informally.

Zuhair, this is the kind of thing I tell you about all the time: You
don't start with a high level, intuitive set of examples of what your
latest proposal for a system (leaving its predecessors by the wayside
in a heap) is attempting to do or represent. And I tell you that it
makes it harder for anyone to contribute and so you get questions
about what you really mean - and you just refer him to someone else's
work. But if this has already been proposed and written about, why
repeat it here? You say it's too time-consuming to explain it, but
spend time dealing with the people who ask you piecemeal what I said
you left out.

C-B