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Topic: What are sets? again
Replies: 21   Last Post: Dec 9, 2012 10:12 AM

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Posts: 1,968
Registered: 12/4/12
Re: What are sets? again
Posted: Dec 4, 2012 5:59 AM
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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


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

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

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


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. ..."


"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

A classical mereology based on the extensionality
of parts would make


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

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

"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

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


requires a criterion for a canonical

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

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

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

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