[FOM] The Gold Standard/correction
Robert M. Solovay
solovay at Math.Berkeley.EDU
Fri Feb 24 02:08:03 EST 2006
Comment at the end of the message:
On Thu, 23 Feb 2006, Harvey Friedman wrote:
> On 2/23/06 4:44 PM, "Robert M. Solovay" <solovay at math.berkeley.edu> wrote:
>
>> ZC is equiconsitent with ZC + Mostowski collapse + "Every set has a
>> transitive closure". The latter theory is much more pleasant for the
>> set-theorist to work in than ZC. {But of course not as nice as ZFC.]
>>
>
> I agree with this. Another way of saying this is to consider the equivalent
> theory to Solovay's:
>
> ZC + (forall x)(therexists an ordinal alpha)(x lies in V(alpha)).
I agree that this theory is nice but it's not equivalent to the
one I gave. One can give a model of the theory I gave [if one uses
Zermelo's version of the natural numbers] which does not contain the
hereditarily finite sets. And Friedman's theory does not entail Mostowski
collapse as the example V(omega + omega) shows.
My favorite weakening of ZFC [but it is considerable stronger than
ZC] is: ZC + Mostowski collapse + V is precisely those sets lying in some
V(alpha) (and for every ordinal alpha V(alpha) exists).
The natural models of this are the V(kappa) for kappa a Beth fixed
point.
--Bob Solovay
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