[FOM] Infinitesimal Calculus

[Ali Enayat]

This is in response to following portion of Harvey Friedman's posting
[May 27, 2009], in which he posed the following question in relation
to the Kanovei-Shelah construction of a definable nonstandard model of
reals:


>The model they construct provably, in ZFC, has cardinality > c. If we
>demand that ZFC also prove that the cardinality is c, then can this be done?

I do not know the answer, but suppose that there is a definable
nonstandard model M of cardinality c. Then, using a well-known
construction, we can associate a unique nonprincipal ultrafilter (on
the natural numbers) with each nonstandard integer of M. This gives
rise to a definable subfamily of the family of all nonprincipal
ultrafilters that has cardinality c. So we are led to a question which
might have a known answer.

Question. Is there, provably in ZFC, a definable subfamily F of
cardinality c of nonprincipal ultrafilters?

Remark. Even if such a family F existed, it is not clear how to modify
the Kanovei-Shelah technique in order to produce the desired model,
since the length of the ultrapower iteration in their construction
would still be more than c,  even if we commit to using all
enumerations of members of F only to specify the iteration "road map".

Regards,

Ali Enayat