# Does $\omega$-consistency of ZF imply $\omega$-consistency of ZFC

Does $$\omega$$-consistency of ZF imply $$\omega$$-consistency of ZFC?

Are there any related results where $$\omega$$-consistency of some intuitionistic logic implies $$\omega$$-consistency of ZFC?

Does this question even make sense to state?

• What notion of $\omega$-consistency are you using? Jan 3 '20 at 17:18

Yes, since the constructible universe $$L$$ is an $$\omega$$-interpretation (interpretation preserving natural numbers) of $$\mathsf{ZFC}$$ in $$\mathsf{ZF}$$. The question does make sense: the $$\omega$$-consistency of a theory $$T\supseteq \mathsf{ZF}$$ (usually) is the assertion that for any formula $$\varphi(x)$$ theory $$T$$ couldn't simultaneously prove $$\exists x\in \omega (\lnot\varphi(x))$$ and all $$\varphi(\underline{n})$$, for individual natural numbers $$n$$. You could formulate $$\omega$$-consistency in the language of first-order arithmetic. And the construction that I have mentioned above in fact shows that over $$\mathsf{PRA}$$ the $$\omega$$-consistencies of $$\mathsf{ZFC}$$ and $$\mathsf{ZF}$$ are equivalent.