Can theories with axioms beyond arithmetic make false promises about integer existence? In this paper, author Nik Weaver warns that there could be questions of $\Sigma_1$-validity of ${\mathrm{ZFC}}$ set theory.
As I understand it, he suggests that the axioms of a set theory might be strong enough to decide PA-undecidable statements about Turing machines, and moreover decide them wrongly. Wrongly in the sense that e.g. ${\mathrm{ZFC}}$ might prove for a particular machine $t$ there to "exist" a $n\in {\mathbb N}$ such that $t$ halts at step $n$ (i.e. ${\mathrm{ZFC}}$ proves $\exists n.T(n)$ for some statement $T$ about a particular coding up of a machine $t$), but "in reality" the machine actually can't do that. This wouldn't be a problem w.r.t. consistency, since it affects a statement where Peano arithmetic (${\mathrm{PA}}$) axioms were too weak to make a call on it.
See here for some relevant definitions and this related SE question. The relation between
${\mathrm{PA}}$ and ${\mathrm{ZFC}}$ is subtle but e.g. from reading the answer on the mentioned questions, one would think ${\mathrm{PA}}$ got us covered for any given Turing machine w.r.t. $\sigma_1$. So I'm trying to understand where a loophole is that Weaver points out.
To clarify, I'm not interested in an inconsistent ${\mathrm{PA}}$ situation or an inconsistent ${\mathrm{ZFC}}$ situation, but I won't presume any soundness properties of the systems. And for the question, I'm also only interested in machines that could (edit: assuming we could wait arbitrary long and provide the memory capacity) be put on the quest to search for a natural and fail to ever halt - despite ${\mathrm{ZFC}}$ claims.
I know that there are conservativity results of ${\mathrm{PA}}$ over ${\mathrm{HA}}$ on, I think, that level of the hierarchy, and similarly with second order-${\mathrm{PA}}$ v. second order-${\mathrm{HA}}$, as well as ${\mathrm{ZF}}$ over ${\mathrm{IZF}}$. I.e. we know that for $\exists n.(f(n)=0)$-statements of ${\mathrm{HA}}$, won't see anything new in ${\mathrm{PA}}$ (${\mathrm{PA}}$ can't fail the world of HA on that level, so to speak). But do we know anything more about ${\mathrm{ZFC}}$ speaking about PA statements?
I'm not sure if results I just listed do in fact clarify the $\Sigma_1$-validity issue on their own. Further (since I read someone bring it up in this context) I wonder whether or how ${\mathrm{ZF}}$ models of itself or of ${\mathrm{PA}}$ could clarify what's possible and what's inconsistent.
Is this issue discussed somewhere on that level of the hierarchy, where validity issues in connection with "the real world" would still be "hands on" in this sense (i.e. not stuff about Turing jumps somewhere removed from recursive machines)? Assuming ${\mathrm{PA}}$ is right about the world, can ${\mathrm{ZFC}}$ be wrong about the world in this "wrong promise/wait forever" kind of way? I do understand that "wrong promise" might be orthogonal to a hard "wrong" notion in proof theory.
 A: The question and comment thread seem a bit all over the place; let me collect the various facts which I think together summarize the situation appropriately.
Any "appropriate" theory $T$ - like $\mathsf{PA}$ or $\mathsf{ZFC}$ - is strong enough to verify all true halting facts. That is, such theories are $\Sigma_1$-complete. Conversely, per Godel no such theory is $\Pi_1$-complete. In particular, if $T$ is "appropriate" in the relevant sense then so is $T+\neg\mathsf{Con}(T)$; since this latter theory is $\Sigma_1$-complete, consistent, and proves a false $\Sigma_1$-sentence, it (and a fortiori $T$ itself) must not be $\Pi_1$-complete.
So a consistent theory of the type we're looking at cannot afford to make any false $\Pi_1$ assertions, but it could conceivably make a false $\Sigma_1$ assertion. This distinction between $\Pi_1$ and $\Sigma_1$ explains why there is no tension between Weaver's observation and the "sufficiency" of $\mathsf{PA}$ (and indeed much less) for $\Sigma_1$ propositions.
Finally, $\mathsf{ZFC}$ could indeed be consistent but $\Sigma_1$-unsound even if $\mathsf{PA}$ is fully sound - at least, working within a reasonably weak background theory. Specifically, assuming $\mathsf{ZFC+Con(ZFC)}$ is consistent in the first place then so is $\mathsf{ZFC+Con(ZFC)+\neg Sound_{\Sigma_1}(ZFC)+Sound(PA)}$. (The "$+\mathsf{Sound(PA)}$"-bit is redundant since $\mathsf{ZFC}$ already proves that $\mathsf{PA}$ is fully sound, I'm just including it for explicitness.) So basically we cannot use the correctness of $\mathsf{PA}$ to justify even the $\Sigma_1$-correctness of $\mathsf{ZFC}$ without baking the latter assumption into our base theory at the outset. However, this happens long before we hit $\mathsf{ZFC}$: the same holds for $\Pi^1_1$-$\mathsf{CA_0}$, if memory serves, which is a tiny fragment of $\mathsf{Z_2}$ which is itself a tiny tiny tiny fragment of $\mathsf{ZFC}$. So leaping all the way to $\mathsf{ZFC}$ here is massive overkill, and I think makes this phenomenon appear more mysterious than it actually is.
