## New answers tagged theories-of-arithmetic

21
votes

### Can we interpret arithmetic in set theory, with exactly PA as the ZFC provable consequences?

The standard terminology is that an interpretation $I$ of a theory $U$ in a theory $T$ is faithful if for all sentences $\phi$ in the language of $U$,
$$T\vdash\phi^I\iff U\vdash\phi.$$
(Here and ...

17
votes

Accepted

### Can we interpret arithmetic in set theory, with exactly PA as the ZFC provable consequences?

This is equivalent to the $\Sigma_1$-soundness of $\mathsf{ZFC}$ (and this equivalence is highly robust to replacing $\mathsf{PA}$ with some other theory):
If $\mathsf{ZFC}$ is $\Sigma_1$-sound then ...

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