Von Neumann gave a clear definition of his universe using ordinal induction
This is highly debatable. There are two rather extreme gaps here
- What are the ordinals?
- What is a power set?
that are arguably not sharp concepts$.^*$
If you nonetheless believe this is clear (and that furthermore it's clear that this object obeys the ZFC axioms), then I agree, that is a valid basis on which to infer the consistency of ZFC.
On the other hand, if you're under the impression that all is well not because of Platonist convictions, but merely because we can define the Von-Neumann universe in axiomatic set theory (or class-set theory), then we have to think clearly about whether we can actually formulate such a consistency proof ("$V$ is a model for ZFC, therefore ZFC is consistent") in the theory we're working in.
In ZFC or weaker set theories, the answer is obviously not, on grounds of the incompleteness theorem, if nothing else. But it goes deeper than that since, although we can define $V$ as a proper class in ZF - foundation, we can't even express "$V$ is a model of ZF". This is because there's no way in general of formulating a satisfaction predicate for proper classes, and in fact, Tarski's theorem implies that it will always fail when $V$ is the whole universe. This even holds through in stronger theories, e.g. ZFC + inaccessible, where we can prove the consistency of ZFC via set-sized models but still can't internally make sense of the whole universe being a model of it.
So given the issues of with the first order approach, we would naturally want to attack this in a class-set theory. As mentioned in Asaf's answer linked in the comments, in NBG, we run into a similar issue as before that we can't construct a first-order satisfaction predicate. Whereas, in Morse-Kelley, we can construct one, and the argument "$V$ is a model for ZFC, therefore ZFC is consistent" goes through just fine there.
So, as with anything, whether we accept that ZFC is consistent is a function of what we're willing to assume.
On the matter of the "logical inference rules" you bring up, there's something interesting we can say. The fact that we can interpret formal proofs as relative to some set or class, and that under reasonable assumptions, they do correctly reflect inference rules, leads to relative consistency proofs. For instance, $V$ can be defined in ZF - foundation, but $V$ satisfies all of ZF. From this, we can deduce that if ZF - foundation is consistent, then ZF is consistent, since any proof of inconsistency from ZF could be carried out in ZF - foundation, relativized to $V.$ Similarly, the constructible universe $L$ can be defined in ZF, but it satisfies ZFC + GCH + Suslin trees exist, etc., establishing a bunch of equiconsitencies with ZF.
But if we just try to do it with $V$ in a ZF(C) background, though, we don't get anything, since we can prove that $V$ is the whole universe, anyway. So we just get the boring result that Con(ZF(C)) implies Con(ZF(C)).
$^*$Since this remark has attracted one perplexed comment and ostensibly a downvote from some other user, let me explain more precisely what I mean here. I don't intend to say that the Von-Neumann universe is not precisely definable in set theory... it is. What I am pushing back on is the implication in the question that the Von Neumann universe construction somehow defines the set theoretical universe.
Sure, it's easy to define what a subset is and the power set axiom guarantees the power set operation exists. And replacement guarantees the ordinals "go on forever" in some reasonable sense, or at least that they don't have an unnatural stopping point. But what the ordinals and power sets precisely are depends in an inpredicative way on an already existing universe of sets. And this is how it's generally taught in set theory books and courses: Von Neumann's hierarchy does not "construct" the universe of well-founded sets from the bottom up, it merely gives a nice stratification of things that were already there.
And I'm not making this assertion that these concepts are arguably vague out of left field (in fact I don't fully agree that they are)... this is a well-known matter of debate. See e.g. here.
