Can $ZFC + \neg Con(ZFC)$ be interpretable in $PA + \omega$- rule?

Question

Suppose ZFC is consistent. Then can $ZFC + \neg Con(ZFC)$ be interpretable in $PA + \omega$- rule?

The idea is that "interpretability" doesn't preserve truth, so even if we hold ZFC + CH to be true, still we can interpret ZFC+ $\neg$ CH within it. Similarly $ZFC + \neg Con(ZFC)$ is a false theory if ZFC is consistent. But for $PA + \omega$-rule to interpret it we must write every axiom of it in the language of arithmetic using some interpreting formula, and then we must of course prove all of those arithmetical interpreting sentences to be true in $PA+\omega$-rule. This would mean that we'd prove an arithmetical interpretation of $\neg Con(ZFC)$ to be true in PA + $\omega$-rule. But that shouldn't be, isn't it. Because what is proved in PA + $\omega$-rule is only true arithmetical sentences, and $\neg Con(ZFC)$ is not true (if ZFC is true). If that is correct, then this would supply a negative answer to the above question. This would mean that every theory $T$ if it finds an interpretation in PA + $\omega$-rule, then $T$ is to be labeled as TRUE.

I'm asking this question because before I was under the impression that any consistent first order theory $T$ would have an arithmetical sentence Con(T) that is provable in PA + $\omega$-rule, and so it can be interpretable in the latter theory (in order for the consistency statement about it to make sense of being about it).

Which one is right? This is really confusing.

Answer 1

Any consistent, recursively axiomatized theory (for example ZFC + $\neg$Con(ZFC)) can be interpreted in true arithmetic, i.e., the first-order theory consisting of all sentences true in the standard model of arithmetic (equivalently, all sentences provable in PA + $\omega$-rule). The proof is essentially the usual Henkin proof of the completeness theorem. By paying attention to the quantifier complexity of the steps in that proof, one gets that such a theory has an arithmetically definable (in fact $\Delta^0_2$) model, and that amounts to an interpretation in true arithmetic.

Your argument for the contrary conclusion confuses "$\neg$Con(ZFC) is true" (which is not the case) with "the interpretation of $\neg$Con(ZFC) is true in the Henkin model constructed above" (which is true because it's a Henkin model for the theory ZFC + $\neg$Con(ZFC)).