🚀 go-pugleaf

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.  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.  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 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.