🚀 go-pugleaf

In article <1992Nov3.201225.9524@CSD-NewsHost.Stanford.EDU> 
pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:

>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:

VP:
>>>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.

RH:
>>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!)

VP:
>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?

ZF proves that every set has a rank.

VP:
>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.

Obviously, you cannot refer to the totality of sets in ZF.  Try
thinking charitably, -- it will aid your comprehension.

VP:
>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?

You can talk about the hierarchy, -- without referring to it as a
completed totality, -- by axiomatizing the notion of rank.  See Dana
Scott's paper "Axiomatizing Set Theory" in the 1974-5 AMS colloquium,
and Michael Potter's elementary book _Set Theory_, based on Scott's
axiomatization.

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

cordially,
mikhail zeleny@husc.harvard.edu
" -- I shall speak bluntly, because life is short."