I would suggest changing this system by defining to mean that no is the Gödel number of a proof of an inconsistency in ZFC (instead of just asserting that isn’t). The purpose of this is to make it so that if ZFC were inconsistent, then we only end up talking about a finite number of levels of truth predicate. More specifically, I’d define to be PA plus the axiom schema

Then, it seems that Jacob Hilton’s proof that the waterfalls are consistent goes through for this waterfall:

Work in ZFC and assume that ZFC is inconsistent. Let be the lowest Gödel number of a proof of an inconsistency. Let be the following model of our language: Start with the standard model of PA; it remains to give interpretations of the truth predicates. If , then is false for all . If , then is true iff is the Gödel number of a true formula involving only for . Then, it’s clear that , and hence all (since is the strongest of the systems) is sound on , and therefore consistent.

Thus, we have proven in ZFC that if ZFC is inconsistent, then is consistent; or equivalently, that if is inconsistent, then ZFC is consistent. Stepping out of ZFC, we can see that if is inconsistent, then ZFC proves this, and therefore in this case ZFC proves its own consistency, implying that it is inconsistent. Hence, if ZFC is consistent, then so is .

(Moreover, we can formalize this reasoning in ZFC. Hence, we can prove in ZFC (i) that if ZFC is inconsistent, then is consistent, and (ii) that if ZFC is consistent, then is consistent. By the law of the excluded middle, ZFC proves that is consistent.)

We should be more careful, though, about what we mean by saying that φ(x) only depends on Trm for m>n, though, since this cannot be a purely syntactic criterion if we allow quantification over the subscript (as I did here). I’m pretty sure that something can be worked out, but I’ll leave it for the moment.