🚀 go-pugleaf

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.