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 EA) 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 (G\"odel 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.