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Assume that $\sf ZFC$ has a transitive model, and let $\varphi$ be a $\Sigma_1$-sentence in the language of set theory. Assume that $\varphi$ holds in every transitive model of $\sf ZFC$. Will it follow that $\varphi$ holds in all models of $\sf ZFC$?

Of course the statement isn't true in general for at least $\Pi_1$ sentences (for example, $\sf Con(ZFC)$), and both conclusions are possible if we don't assume the existence of a transitive model, but what is the answer in this setting? Thank you.

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    $\begingroup$ By "$\Sigma_1$" do you mean $\Sigma_1$ in the Levy hierarchy or $\Sigma^0_1$ in the arithmetical hierarchy? ($\mathsf{Con(ZFC)}$ is in fact $\Delta_1$ in the Levy hierarchy.) $\endgroup$
    – Noah Schweber
    Sep 20 at 23:50
  • $\begingroup$ @NoahSchweber I meant Lévy hierarchy, so that answers my question, thank you. $\endgroup$
    – Tan
    Sep 20 at 23:58
  • $\begingroup$ The answer for the arithmetical hierarchy is “yes”, incidentally. $\endgroup$
    – spaceisdarkgreen
    Sep 21 at 0:15

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In fact, every arithmetical sentence $\varphi$ is $\Delta_1$ in the Levy hierarchy:

Some/every structure satisfying [basic properties of $(\omega;+,\times)$] satisfies $\varphi$.

The point is that "satisfies $\varphi$" winds up using only bounded quantifiers in the set-theoretic sense (e.g. "$\forall x\in\omega$"). This may be easier to see if instead of the semiring of natural numbers we think about the set $L_\omega$, also called $V_\omega$ or $\mathsf{HF}$.

In particular, $\mathsf{Con(ZFC)}$ is a $\Delta_1$ sentence true in every transitive model of $\mathsf{ZFC}$ but not (hopefully!) $\mathsf{ZFC}$-provable.

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