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fom

Posts: 1,969
Registered: 12/4/12

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

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Recently, Zuhair made a post concerning the
consistency of Zermelo-Fraenkel set theory
with the axiom of choice relative to
Morse-Kelley set theory. This would not surprise
me at all, although Charlie-boo'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,
Morse-Kelley 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 set-theoretic, 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 upward-directed 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 Zermelo-Fraenkel 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 n-element
set S (or an n-set 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
n-set S such that

-((A_i n A_j)=null)

for all A_i, A_j in A, then |A|<=2^(n-1). Further,
if |A|<2^(n-1), A can be extended to a collection
of 2^(n-1) 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'=(S-A) is
certainly not in A, since

(A_k n A_k')=null

So, we immediately obtain

|A|<=(1/2)*2^n=2^(n-1)

This bound cannot be improved upon since the
collection of all subsets of {1,...,n}
containing 1 satisfies the intersection
condition and has 2^(n-1) members.

Now suppose

|A|<=2^(n-1)

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^(n-1) members, repeat the
process.

============

The first thing to observe in this proof is the
assumption that

|A|<=2^(n-1)

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 Zermelo-Fraenkel
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)
non-empty sets with the empty set as a member

3)
non-empty 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
well-orderable 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
"second-order" theory, note that in second-order
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 Morse-Kelley set theory.
Morse-Kelley set theory as presented in Kelley
presumes a global axiom of choice.