🚀 go-pugleaf

In article <1992Nov3.041515.27732@CSD-NewsHost.Stanford.EDU> pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:
>In article <1992Nov2.205433.21208@tamsun.tamu.edu> cmenzel@tamsun.tamu.edu (Chris Menzel) writes:
>>In article <1992Nov2.040843.4524@CSD-NewsHost.Stanford.EDU> pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:
>>>That it [Foundation Axiom] is self-evident to you now seems more
>>>an accident of culture---your training---than of something inherent in
>>>the notion of set....
>>
>>I think that's too strong if, like me, you were introduced to set
>>theory via the iterative hierarchy.  If that's your picture of sets,
>>then the axiom *is* self-evident, and well-foundedness *is* an
>>inherent property of sets.  The question is rather whether the
>>iterative hierarchy encompasses *all* the sets, or whether we've just
>>been looking at a rather limited portion of the universe, as is the
>>case if we take the universe to be nonwellfounded ala Aczel.
>
>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.
>
>Anyone brought up on the iterative hierarchy was told a lie.
>-- 
>Vaughan Pratt			Ain't no truth in logic, son.

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!)


-- 
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