[FOM] The Gold Standard/correction

[Harvey Friedman]

On 2/23/06 12:45 AM, "Robert M. Solovay" <solovay at math.berkeley.edu> wrote:

> On Wed, 22 Feb 2006, Harvey Friedman wrote:
> 
>> 
>> They will instantly recognize all of the axioms of ZFC except replacement
>> and foundation, if you gently remind them. (Replacement and foundation are
>> rarely used, and ZFC\Rep\Found is of course equiconsistent and mutually
>> interpretable with ZFC).
> 
> This seems like a howler. The theory ZFC - Replacement is
> essentially Zermelo which has considerably less consistency strength than
> ZFC.

Yes, a howler. I should have written:

They will instantly recognize all of the axioms of ZFC except replacement
and foundation, if you gently remind them. (Replacement and foundation are
rarely used, and ZFC\Rep\Found is of course equiconsistent and mutually
interpretable with Z and ZC).

ZC looks like the closest formalization of general mathematical practice -
IF one insists on no coding.

Of course, some interesting stuff is missing...

This assessment is based on the premise that Grothendieck universes are to
be properly viewed as a superfluous aberration. If not, then probably NBG +
AxC would be the preferred vehicle, under the no coding rule.

Harvey Friedman