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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 ...
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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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