Ultimate Truth vis-à-vis stable Truth

We show that the set of ultimately true sentences in Hartry Field’s Revenge-immune solution model to the semantic paradoxes is recursively isomorphic to the set of stably true sentences obtained in Hans Herzberger’s revision sequence starting from the null hypothesis. We further remark that this shows that a substantial subsystem of second order number theory is needed to establish the semantic values of sentences in Field’s relative consistency proof of his theory over the ground model of the standard natural numbers: ∆3-CA0 (second order number theory with a ∆3-Comprehension Axiom scheme) is insufficient. We briefly consider his claim to have produced a “revenge-immune” solution to the semantic paradoxes by introducing this conditional. We remark that the notion of a “determinately true” operator can be introduced in other settings.