🚀 go-pugleaf

In article <1992Nov4.015603.16555@CSD-NewsHost.Stanford.EDU> pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:
>In article <1992Nov3.201225.9524@CSD-NewsHost.Stanford.EDU> pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:
>>
>>Or do you mean that in ZF one can prove that every set can be built up
>>iteratively?  Assuming the Foundation Axiom I have no quarrel with the
>>latter, but then you just have a statement about individual sets being
>>built up iteratively, not about their collectively forming a
>>hierarchy.
>>
>>I'm probably just being my usual dense self, but I just can't figure
>>out how to talk about the hierarchy itself within ZF.  What is the
>>trick?  And what is a more precise wording of the theorem you refer to
>>above?
>
>On reflection I guess the best answer to the second question is bound
>up in the first: the hierarchy does exist (assuming FA), not in the
>sense that you can actually say explicitly in ZF that it exists, but
>rather as a sort of metastatement whose translation into von
>Neumann-Bernays set theory would allow the hierarchy to exist as a
>class.
>
>This leaves open the question of existence of the cumulative
>hierarchy.  We can't say it exists the way we can say that natural
>numbers exist, or that the set of natural numbers exists.  However it
>acquires a certain meta-existence by virtue of a suitable
>interpretation of ZF formulae.
>
>So to say that the existence of the cumulative hierarchy is an outright
>lie is too strong.  Whether to call it a white lie or a gray truth
>seems the more appropriate question.
>
>The discussion thus far has all been predicated on the consistency of
>ZF.  The outcome I would naturally prefer is that I win my $100 bet
>that ZF is inconsistent.  That would make it a lie for sure.
>-- 
>Vaughan Pratt				There's no truth in logic, son.

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 iterative hierarchy is a perfectly sound way of
describing the "shape" of the "universe" of ZFC.

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).


-- 
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