Question

I want to show the following theorem from Feferman and Kreisel:

Let $\phi$ be such that there is a theorem $\sigma$ of ZFC so that for every transitive model $M$ of $\sigma$, we have: $\phi^M \to \phi$.

Then: $\phi$ is equivalent to a $\Sigma_1$ formula.

Can someone give me a proof or some bibliographic references please?

Answer 1

Suppose that $\phi$ is an assertion and there is a sentence $\sigma$ provable in ZFC such that whenever $\sigma$ holds in a transitive set $M$ and $\phi$ also holds in $M$, then $\phi$ is true.

Since there definitely are transitive sets in which $\sigma$ is true, by the Levy reflection theorem, it follows that $\phi$ is true provided that there is a transitive set $M$ such that $\sigma$ and $\phi$ are both true in $M$. And conversely, if $\phi$ is true, then again by reflection there are transitive sets in which both $\sigma$ and $\phi$ are true. Thus, $\phi$ is equivalent to the assertion: "There is a set $M$ such that $M$ is transitive and $\langle M,{\in}\rangle\models\sigma\wedge\phi$". But this is a $\Sigma_1$ assertion in set theory, because you have the initial unbounded quantifier $\exists M$, followed by assertions all of whose quantifiers may be bounded by $M$. Thus, the only unbounded quantifier is $\exists M$, and so $\phi$ is equivalent to a $\Sigma_1$ assertion, as requested.

I am sorry that I don't know any bibliographic reference.