[FOM] Arithmetical soundness of ZFC

[laureano luna]

Nik Weaver wrote:

>Is every sentence of first order
>number theory that is provable in ZFC actually true?

>(...)

>I have the impression that it is fairly common for
>philosophical defenses of set theory to miss this point, and
>to assume that the only issue is the question of consistency.

I wonder how this could be the case after Gödel. 

Incompleteness entails the existence of many consistent but mutually incompatible arithmetical extensions of any arithmetical system. I'd say this compels to distinguish soundness from consistency.

Regards,

Laureano Luna