🚀 go-pugleaf

In article <1992Nov3.162547.25630@guinness.idbsu.edu> holmes@garnet.idbsu.edu (Randall Holmes) writes:
>In article <1992Nov3.041515.27732@CSD-NewsHost.Stanford.EDU> pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:
>>The iterative hierarchy is consistent with Z but not with ZF.
>>Replacement (F) says in effect that any construction you can iterate
>>yields a set.  Hence if the universe is an iterative hierarchy it is a
>>set.
>>
>
>Same remark as before (this is ludicrous) + the obvious remark that
>set-theorists will be awfully surprised by this pronouncement, since
>they use the iterative hierarchy all the time (iterated power sets of
>the empty set indexed by the ordinals, taking unions at limit
>ordinals, which exhaust the universe in ZF -- the iterative hierarchy
>is not only consistent with ZF, it is established by a THEOREM of ZF!)

What exactly do you mean by "ZF can establish the iterative
hierarchy"?  Do you mean that it can establish the *existence* of the
hierarchy?  Why wouldn't that be equivalent to asserting the
consistency of ZF within itself?

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?
-- 
Vaughan Pratt			Ain't no truth in logic, son.