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fom
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CON(ZF) and the ontology of ZF
Posted:
Feb 17, 2013 3:42 AM


Recently, Zuhair made a post concerning the consistency of ZermeloFraenkel set theory with the axiom of choice relative to MorseKelley set theory. This would not surprise me at all, although Charlieboo's remark to make the formal statements available applies.
There are two things in Zuhair's post that make me believe that he may have succeeded. First, MorseKelley set theory has a global axiom of choice. Secondly, Zuhair has introduced a size limitation relative to transitive closures.
What is actually involved in what Zuhair has done, for those who would like to look at something not settheoretic, is a closure algebra.
http://en.wikipedia.org/wiki/Closure_algebra
Transitive closure is a closure operation and in his papers on algebraic logic, Paul Halmos describes existential quantification in terms of a mapping E of a Boolean algebra A into itself such that
1) (E0)=0
2) p<=(Ep)
3) E(p \/ q)=(Ep \/ Eq)
4) E(Ep)=(Ep)
5) E(Ep)'= (Ep)'
As noted by Halmos, the first four axioms are correlated with Kuratowski's closure axioms:
1) cl(null)=null
2) Ax(x<=cl(x))
3) AxAy(cl(x u y)=(cl(x) u cl(y))
4) Ax(cl(cl(x))=cl(x))
Within set theory, transitive closures follow from von Neumann's axiom of regularity. In comparison with the systems above,
1) TC(null)=null
2) Ax(x<=TC(x))
3) AxAy(TC(x u y)=(TC(x) u TC(y)))
4) Ax(TC(TC(x))=TC(x))
So, given Zuhair's relevant statements,
Definition: x << y iff x < y & for all z in TC(x). z < y
Hereditary size limitation: set(x) <> Exist y. set(y) & for all m in x (m << y)
one has
1) every element of x is a subset of y
2) every element of the transitive closures for the elements of x are subsets of y
3) set(x) is true iff such a y exists and set(y) is true.
If these paraphrases are correct interpretations, then the third axiom is reflecting the axiomatically asserted existence of any set upward. In other words, his axiom is describing the third axiom one would associate with a directed class structure.
The axioms for a directed set in are given as
1) Ax(x>=x)
2) AxAyAz((x>=y /\ y>=z) > x>=z)
3) AxAyEz(z>=x /\ z>=y)
Whether or not Zuhair's other axioms manage to satisfy the first two axioms is not of concern here. And, the third axiom holds for the elements of x. I do not immediately see that they hold for x itself in relation to y.
But, it is important to recognize that the directionality given by his axioms are precisely that used to construct forcing languages.
In the WMytheology threads this directed set structure also seems fundamental to the finitism debate. Although I could be wrong, it seems to me that this upwarddirected directed set structure is precisely at the heart of Euclid's proof of infinitely many prime numbers. So, Zuhair seems to have captured a fundamental expression of succession without the elaborate mechanisms of forcing.
And, of course, not only does the axiom of regularity ground the existence of transitive closures, it grounds the mechanism of transfinite recursion.
In transition from Zuhair's axiom of size limitation, note that he always assumes the usual axioms for identity:
Language: FOL(=,in)
In the post,
news://news.giganews.com:119/88qdnU1ZNsB9j4LMnZ2dnUVZ_uudnZ2d@giganews.com
I discuss careful usage like this. After writing that post, I realized that I could say more and speak directly to the ontology of ZF.
The ontological assumption placed upon (but not necessarily intrinsic to) the set theory described in Zermelo 1908 may be formalized as follows:
1) Ax(x=x)
2) AxAy(x=y <> Ez(x=z /\ z=y))
3) ExAy((yex <> y=x))
4) Ax(x=V() <> Ay((yex <> y=x)))
5) AxAy(Az(zex /\ zey) > x=y)
The second axiom was introduced by Tarski and Monk in their work on cylindrical algebras,
http://en.wikipedia.org/wiki/Cylindric_algebra
However, to obtain the converse to the axiom of extension, one must respect Zermelo's original intent of introducing a null class and singletons. And, with the definite description of a universal class the axioms must be recast  much like Zuhair's work  so that a criterion for excluding the universal class from membership is in place. So, the usual notion of pairing needs to be altered to something along the lines of
AxAy((Ez(xez) /\ Ez(yez)) > EvEw(Az(zew > (z=x \/ z=y)) /\ wev))
Now, as Zermelo observed, the converse to the axiom of extension is obtained mereologically relative to the singletons, themselves individuated relative to the null class.
One can get the sense of this from Euclid:
Book VII, Definition I:
A unit is that by virtue of which each of the things that exist is called one.
Book I, Definition I:
A point is that which has no part.
So, a null class has no parts, and singletons are individual by virtue of the ontological invariance that the received paradigm attaches to the sign of equality.
Within set theory, singletons are the atomic parts. But, they cannot be mereologically individuated by a formal statement such as
AxAy(x=y <> Az(zcx <> zcy))
where the subset relation is strict. In order for other sets to have this property, there must be a null class to act as a halt to mereological regress.
One can compare the situation with the semiotics of Pierce and Saussere. Saussere uses "observable sign vehicles"  basically inscriptions  to halt the semiotic regress. Pierce has no such halt, and, like some of what is found in Carnap, the notion of sign and significance regress into an infinity of abstraction.
As one wishes to stay away from this kind of thing, it is time to turn to a finitary analogue to the kind of entanglement that distinguishes the nature of ZermeloFraenkel set theory from naive set theory.
The very first example from "Combinatorics of Finite Sets" by Ian Anderson proceeds as follows:
============
Problem.
Let A be a collection of subsets of an nelement set S (or an nset S) such that
((A_i n A_j)=null)
for each pair (i,j). How big can A be? The answer, and more besides, is given by the following theorem.
Theorem 1.1.1
If A is a collection of distinct subsets of the nset S such that
((A_i n A_j)=null)
for all A_i, A_j in A, then A<=2^(n1). Further, if A<2^(n1), A can be extended to a collection of 2^(n1) subsets also satisfying the given intersection property.
============
Before examining the proof, consider Lesniewski's criticism of the Russellian analysis and its paradox. What is of particular note here is Lesniewski's concept of a full class. He writes:
"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 the proof from Ian Anderson's book, the thing to consider is the role played by complements and how that role relates to the situation where the subclasses of the problem definition are extended.
============
Proof
If A_k in A, then the complement A_k'=(SA) is certainly not in A, since
(A_k n A_k')=null
So, we immediately obtain
A<=(1/2)*2^n=2^(n1)
This bound cannot be improved upon since the collection of all subsets of {1,...,n} containing 1 satisfies the intersection condition and has 2^(n1) members.
Now suppose
A<=2^(n1)
Then there must be a subset A_k with
(A_k in A)
and also
(A_k' in A)
We then add A_k to the collection A unless there exists B in A such that
(A_k n B)=null
But, then B is a subset of A_k' and so we could add A_k' to A. If the resulting collection has fewer than 2^(n1) members, repeat the process.
============
The first thing to observe in this proof is the assumption that
A<=2^(n1)
This is precisely the assumption made for the generation of forcing models in set theory. It is a correct assumption for the purpose of investigating the relative consistency of various statements, just as had been done historically with the parallel postulate.
But, foundationally, such an assumption is disastrous and only makes sense in the context of the confusion brought upon by treating Russell's paradox as an ontological necessity. Tarski's axiom,
2) AxAy(x=y <> Ez(x=z /\ z=y))
now admits "epistemic informative identity" into a formal ontology for ZermeloFraenkel set theory. Understanding what it means for a class to be "full" in the sense suggested by Lesniewski is not hard. The necessary condition is the "almost universality" which characterizes the constructible universe.
Almost universality holds that every part of the set universe is captured as a part of the cumulative hierarchy. Consequently, it is captured as an element at the next iteration of the hierarchy. Its complement is excluded and it is "full" in the sense of what is described by Anderson's proof.
One of Cantor's principles had been that "finished" sets be as similar as possible to finite sets. Anderson's example provides a finitary example of how Zermelo's principle established a set theory that did not have the contradictions of naive set theory.
It also provides a means by which to analyze von Neumann's axiom of foundation.
The axiom of foundation distinguishes three classes of sets. Namely,
1) the empty set
2) nonempty sets with the empty set as a member
3) nonempty sets not having the empty set as a member
According to Anderson's example, the cardinality of the given collection, when described in terms of the filterbase about a point cannot be improved upon.
The cumulative hierarchies made possible by von Neumann's axiom of foundation are indexed by the ordinals. The equinumerosity of infinite multiplicities with their submultiplicities should not blind us to the fact that Zermelo's construction forces us to accept almost universal wellorderable models as the only meaningful candidates for faithful models of set theory.
At the link,
http://en.wikipedia.org/wiki/Freiling%27s_axiom_of_symmetry
one can find explanations for the remark:
"The above can be made more precise:
ZF  (AC_(P(kappa)) <> AX_(kappa)) <> CH_(kappa)
This shows (together the fact that the continuum hypothesis is independent of choice) a precise way in which the (generalised) continuum hypothesis is an extension of the axiom of choice."
Since mereology is generally viewed as a "secondorder" theory, note that in secondorder logic,
L=HOD
So, returning to the statements in the opening paragraph, it does not surprise me that Zuhair may have succeeded in devising a means by which to show Con(ZF) relative to MorseKelley set theory. MorseKelley set theory as presented in Kelley presumes a global axiom of choice.



