From: holmes@garnet.idbsu.edu (Randall Holmes)
Subject: Re: Impredicativity - was: Russell's Paradox.
Organization: Boise State University
Date: Thu, 5 Nov 1992 16:33:56 GMT

In article <1992Nov4.221041.4812@CSD-NewsHost.Stanford.EDU> pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:
>In article <1992Nov4.161936.12444@guinness.idbsu.edu> holmes@garnet.idbsu.edu (Randall Holmes) writes:
>>
>>The iterative hierarchy does _not_ exist as a completed totality, but
>>saying that it "does not exist" is like saying that the ordinals "do
>>not exist".
>>
>>The theorem I have in mind is "for any set x, for some ordinal a, x E
>>V(a)", where V(a) is defined by
>>
>>                V(0) = {}
>>                V(a+1) = P{V{a}}
>>                V(L) = U{a < L}[V(a)] (L a limit ordinal).
>>
>
>That is, every set has an ordinal rank, a theorem that depends on FA.
>I think we're in complete agreement that this statement is ZF+FA's way
>of asserting the existence of the cumulative hierarchy.  That should
>put the situation in sufficiently simple terms that people can judge
>for themselves the relationship of this assertion to the more mundane
>and clearly ZF-expressible existences of ordinary mathematical
>objects.  The latter kind of existence (a) is made honestly with
>quantifiers and (b) does not depend on FA.  Decide for yourself how you
>feel about the former kind.
>
>Point (a) is the weaker of the arguments, since we can simply translate
>our framework to von Neumann-Bernays set theory, where we really can
>state that the *class* of sets exists, as does the class of ordinals.
>In a way this sharpens the issue even better, by reducing a question
>about quantifiers to one about classes vs. sets.
>
>By way of at least justifying my position, if not persuading you of it,
>let me focus on (b).
>
>That the theorem does depend on FA means that from the perspective of
>those repudiating FA, there may well be no cumulative hierarchy, in the
>sense that there may well exist sets outside any such hierarchy.

This is quite correct.  If Choice and Foundation are not available,
there is no basis for constructing the iterative hierarchy.
Alternatively, one could work in ZFA with a proper class of atoms.


  Such
>people will have no quarrel with my "Anyone brought up on the iterative
>hierarchy was told a lie".  At the time I uttered this inflammatory
>heresy I was mounting a defense of AFA, in which context this can be
>seen to be a perfectly valid position, since as we have seen, refuting
>my heresy entails accepting FA.

This is irrelevant to AFA.  The universe of AFA _is_ hierarchical, in
a way which follows the structure of the universe of ZFC slavishly.
See another recent post of mine.

  Attacking Aczel is like attacking
>Brouwer, with the difference being that, in view of how English (and I
>imagine many other languages) assigns significance to double negatives
>and does not exclude the middle, there would appear to be more
>linguistic support for Brouwer than Aczel can reasonably hope for.  On
>the other hand there is at least support from some circles of computer
>science for dropping FA in order to allow membership cycles.
>
>It should be pointed out that neither of the two axiomatizations of ZF
>(that I took as definitive of ZF for my bet that ZF will be shown
>inconsistent by 2012) mention FA.  These were the axiomatizations of
>Takeuti and Zaring in their book "Introduction to Axiomatic Set
>Theory", and of Schoenfield in his article in Barwise's "Handbook of
>Mathematical Logic."  I selected these only because of their
>accessibility, not because they omitted FA.  In fact I do not know of
>an equally accessible axiomatization of ZF that includes FA (we
>amateurs are appallingly ignorant).  Randall, you're the one appealing
>to FA here, can you suggest a suitably accessible axiomatization that
>includes it?  It would come in handy for future reference.

I can't think of an especially "accessible" one.  I have Thomas Jech,
_Set Theory_, and Kunen, _Set Theory_, on my shelf, both of which list
the axioms, but both of which require some sophistication to
understand the axioms as given.  Azriel Levy's book on Sets and
Classes is a good source for alternative axiomatizations of set
theory.


>
>I should state again, for the benefit of people wanting to bet on the
>matter, that the consistency of ZF is unaffected by whether any or all
>of AFA, FA, AC, or CH is added.  But dropping any one of F
>(Replacement), the Power Set axiom, or the Axiom of Infinity, makes a
>*big* difference: I will bet only *against* inconsistency of any of
>those systems, and at very favorable odds to you if you figure these
>have even Job's chance of being inconsistent.  (Job was famous for
>doing his darndest to remain consistent in the face of three inconstant
>friends and a disturbingly inconsistent God, who had been successfully
>tempted by Satan to tempt Job into inconsistency.)
>-- 
>Vaughan Pratt				There's no truth in logic, son.


-- 
The opinions expressed		|     --Sincerely,
above are not the "official"	|     M. Randall Holmes
opinions of any person		|     Math. Dept., Boise State Univ.
or institution.			|     holmes@opal.idbsu.edu
Newsgroups: sci.logic