fom
Posts:
1,968
Registered:
12/4/12


Re: What are sets? again
Posted:
Dec 4, 2012 5:59 AM


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. Parthood: 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)
AntiSymmetry 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 pretheoretic version of classical settheoretic 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 nonselfidenticals
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 selfidentity 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 wellordering 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 urelements 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 > Urelements 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 uratom, being an atom, is a collection of atoms a uratom 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

