# Are $\Sigma_1$-sentences which are true in any transitive model of $\sf ZFC$ necessarily theorems of $\sf ZFC$?

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.

• 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.) Sep 20 at 23:50
• @NoahSchweber I meant Lévy hierarchy, so that answers my question, thank you.
– Tan
Sep 20 at 23:58
• The answer for the arithmetical hierarchy is “yes”, incidentally. Sep 21 at 0:15

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.