Maximal consistent sets of instances of Tarski's schema (T)
As is well known, Tarski's schema \[ \text{Tr }\ulcorner s\urcorner \text{ if and only if } s \tag{T} \] for the truth predicate Tr of sentences \(s\), implies that there can be no theory which contains its own truth predicate satisfying (T). A meta-theory is required. Under circumstances where one wants to develop a semantic theory of a language within that very language, one may deal with Tr as a partial predicate, thereby cutting down on (T). The strategy which the author studies, is to look for maximal sets of sentences which do satisfy (T) and are consistent with the basic laws of arithmetic. He shows that such maximal sets of instances of (T) do exist, although beyond axiomatization. The author concludes: ``Thus the requirement that our theory of truth should entail a maximal consistent set of instances of (T) is both too weak and too strong. It is too strong because we cannot hope actually to construct a theory that meets the requirement. It is too weak because \dots even if, by some superhuman agency, we got hold of a theory that meet the requirement, we would have no reason to suppose that the theory captured any of the useful features of our ordinary notion of truth''.
scientific article; zbMATH DE number 92833