The title sounds a bit philosophical, but it is

about mathematics. Let me explain.

Consider a first order theory $T$, which is an extension of Peano Arithmetic. Call this theory "good" if it is consistent and satisfies the following

*Property*:
For any $\phi\in\Sigma^0_{n+1}$ such that
$T\vdash \phi$ there exists $\psi\in\Pi^0_{n}$ such that
$T\vdash\psi$ and $PA\vdash\psi\to\phi$.

*Question 1*. Is $ZFC$ "good"?

*Question 2*. The same for $ZFC+something$ (from the lot of proposed new axioms).

*Motivation*.

If $ZFC$ is not "good" then there (EDIT) may be theorems which can be proved in $ZFC$ despite they are false (in the standard model of PA). I believe that $ZFC$ is "good". However, I would like to know if there is a formal proof. (Admittedly, I don't have a slightest idea what a proof may be like). By the way, "goodness" implies consistency, hence a proper proof requires some new axioms (a large cardinal, perhaps).

(EDIT). As Andreas Blass pointed out correctly, even if a theory is not "good" in the above sense, it does not yet follow that some of the theorems are wrong (an obvious fact which I have missed somehow). Still, the question if ZFC is "good" may be of some interest, in my opinion.

*Question 3*. Is "goodness" equivalent to consistency?
(I doubt this).

EDIT: (Clarification). In this question, the theory $T$ is supposed to be "good" and at least as strong as $ZFC$. (Thus, the answer to Question 1 must be yes). The question is, whether $T\vdash Con(T)\to Good(T)$, where $Good(T)$ is a formalization of "goodness"; note that $Good(T)\in \Pi^0_{2}$.

P.S. Is there a standard term for "good"?