[FOM] ZFC and Recursive Inseparability

[Ali Enayat]

This is a reply to a recent query of Santiago Bazerque (Wed, March 9), who 
has asked:

>Does anybody know if ZFC is effectively inseparable? I find this
>question interesting because if it were, then if ZFC + { any
>axiomatizable extension } is consistent, it is recursively isomorphic to
>ZFC (i.e. they would share the same deductive structure) by a result of
>Pour-El and Kripke (Fund. Math. LXI p. 142).

The answer is positive, since ZFC "contains enough arithmetic".  Here 
"enough arithmetic" can be even taken as a finite fragment of Peano 
arithmetic, such as the theory known as Robinson's Q.

Regards,

Ali Enayat