Varieties of truth definitions

Abstract

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 \(\alpha \) which extends a weak arithmetical theory (which we take to be \({{\,\mathrm{I\Delta _{0}+\exp }\,}}\)) such that for some formula \(\Theta \) and any arithmetical sentence \(\varphi \), \(\Theta (\ulcorner \varphi \urcorner )\equiv \varphi \) is provable in \(\alpha \). We say that a sentence \(\beta \) is definable in a sentence \(\alpha \), if there exists an unrelativized translation from the language of \(\beta \) to the language of \(\alpha \) which is identity on the arithmetical symbols and such that the translation of \(\beta \) is provable in \(\alpha \). 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 \(\Sigma _2\)-definable in the standard model of arithmetic. We conclude by remarking that no \(\Sigma _2\)-sentence, satisfying certain further natural conditions, can be a definition of truth for the language of arithmetic.