Varieties of truth definitions

We study the structure of the partial order induced by the definability relation on definitions of truth for the language of arithmetic. Formally, a definition of truth is any sentence α which extends a weak arithmetical theory (which we take to be IΔ _0+exp ) such that for some formula Θ and any arithmetical sentence φ , Θ (⌜φ⌝ )≡φ is provable in α . We say that a sentence β is definable in a sentence α , if there exists an unrelativized translation from the language of β to the language of α which is identity on the arithmetical symbols and such that the translation of β is provable in α . Our main result is that the structure consisting of truth definitions which are conservative over the basic arithmetical theory forms a countable universal distributive lattice. Additionally, we generalize the result of Pakhomov and Visser showing that the set of (Gödel codes of) definitions of truth is not Σ _2 -definable in the standard model of arithmetic. We conclude by remarking that no Σ _2 -sentence, satisfying certain further natural conditions, can be a definition of truth for the language of arithmetic.