I would suggest changing this system by defining ψ(n) to mean that no m≤n is the Gödel number of a proof of an inconsistency in ZFC (instead of just asserting that n 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 Tn to be PA plus the axiom schema
∀m≥n.ψ(m)→∀x.Trm(┌φ(¯¯¯x)┐)↔φ(x).
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 n be the lowest Gödel number of a proof of an inconsistency. Let M be the following model of our language: Start with the standard model of PA; it remains to give interpretations of the truth predicates. If m≥n, then Trm(k) is false for all k. If m<n, then Trm(k) is true iff k is the Gödel number of a true formula involving only Trm′ for m′>m. Then, it’s clear that T0, and hence all Tm (since T0 is the strongest of the systems) is sound on M, and therefore consistent.
Thus, we have proven in ZFC that if ZFC is inconsistent, then T0 is consistent; or equivalently, that if T0 is inconsistent, then ZFC is consistent. Stepping out of ZFC, we can see that if T0 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 T0.
(Moreover, we can formalize this reasoning in ZFC. Hence, we can prove in ZFC (i) that if ZFC is inconsistent, then T0 is consistent, and (ii) that if ZFC is consistent, then T0 is consistent. By the law of the excluded middle, ZFC proves that T0 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.
I would suggest changing this system by defining ψ(n) to mean that no m≤n is the Gödel number of a proof of an inconsistency in ZFC (instead of just asserting that n 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 Tn to be PA plus the axiom schema
∀m≥n.ψ(m)→∀x.Trm(┌φ(¯¯¯x)┐)↔φ(x).
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 n be the lowest Gödel number of a proof of an inconsistency. Let M be the following model of our language: Start with the standard model of PA; it remains to give interpretations of the truth predicates. If m≥n, then Trm(k) is false for all k. If m<n, then Trm(k) is true iff k is the Gödel number of a true formula involving only Trm′ for m′>m. Then, it’s clear that T0, and hence all Tm (since T0 is the strongest of the systems) is sound on M, and therefore consistent.
Thus, we have proven in ZFC that if ZFC is inconsistent, then T0 is consistent; or equivalently, that if T0 is inconsistent, then ZFC is consistent. Stepping out of ZFC, we can see that if T0 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 T0.
(Moreover, we can formalize this reasoning in ZFC. Hence, we can prove in ZFC (i) that if ZFC is inconsistent, then T0 is consistent, and (ii) that if ZFC is consistent, then T0 is consistent. By the law of the excluded middle, ZFC proves that T0 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.