Sentence $\varphi$ of set theory that is satisfied by all well-founded models of ZFC, but which is not a theorem of ZFC.

Question

I think I read somewhere the following.

If a first-order sentence $\varphi$ in the language of set theory holds for every well-founded model of ZFC, then nonetheless:

• $\varphi$ may fail for a non-well-founded model;
• in other words, $\varphi$ needn't be a theorem of ZFC.

What is an example of such a $\varphi$?

Answer 1

Every statement which is in its essence a true [first-order] number theoretic statement in the universe must be true for every well-founded model. The most striking example for these statements are consistency of various theories.$\DeclareMathOperator{\con}{con}$

For example, if there are well-founded models of $\sf ZFC$, then $\con\sf(ZFC)$ holds. It follows that every well-founded model satisfies $\con\sf(ZFC)$. Similarly if there is a model with an inaccessible cardinal, then $\con\sf(ZFC+I)$ holds, so it must hold in every well-founded model, and if there is a model with a proper class of supercompact cardinals, then in every well-founded model it is true that there is a model with a proper class of supercompact cardinals.

On the other hand, if there is a model of $\sf ZFC$ then there is a model for $\sf ZFC+\lnot\con(ZFC)$. And this model is necessarily not well-founded, and in fact not even an $\omega$-model (meaning: it has non-standard integers).