Of course, we can also investigate theories which result by adding the truth-theoretic axioms to much stronger theories like set theory. Usually there is no chance of proving the consistency of set theory plus further truth-theoretic axioms because the consistency of set theory itself cannot be established without assumptions transcending set theory. In many cases not even relative consistency proofs are feasible. However, if adding certain truth-theoretic axioms to PA yields a consistent theory, it seems at least plausible that adding analogous axioms to set theory will not lead to an inconsistency. Therefore, the hope is that research on theories of truth over PA will give an some indication of what will happen when we extend stronger theories with axioms for the truth predicate. However, Fujimoto (2012)  has shown that some axiomatic truth theories over set theory differ from their counterparts over Peano arithmetic in some aspects.