[FOM] The Gold Standard/correction

[Harvey Friedman]

On 2/24/06 2:08 AM, "Robert M. Solovay" <solovay at math.berkeley.edu> wrote:

Solovay 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.]
>>> 

Friedman wrote:
 
>> 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)).
>
Solovay wrote:

> 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.
> 

I was thinking of the weak Mostowski collapsing theorem in Jech's set theory
book (vol. 2), p. 88, where he also mentions the strong form. Of course you
are referring to the strong form, which asserts that an extensional well
founded relation is isomorphic to a transitive set under epsilon.

So I stand corrected.

Also, I now recall that Zermelo set theory does not prove the existence of
V(omega), or even the usual form of infinity we use today with x union {y}.

Readers may be interested in the exact theory in Zermelo's article, 1908,
Investigations in the foundations of set theory I, translated, in von
Heijenoort. I was surprised to find the axiom of choice plus this
formulation of infinity: there exists A such that A has the empty set and is
closed under the singleton operation.

Specifically, 
1. Extensionality.
2. Empty set plus pairing.
3. Separation.
4. Power set.
5. Union.
6. Choice.
7. Infinity. (as above).

Note: no foundation and no replacement. Also, empty set follows immediately
from 3, so axiom 2 can be taken to be just pairing.

Because of the interest then and now in issues surrounding choice, it is
reasonable to let Z stand for 1-7 without choice, and ZC stand for 1-7.

Harvey Friedman