Varieties of truth definitions
Piotr Gruza, Mateusz Łełyk · arXiv (Cornell University) · 2023
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 EA) such that for some formula $Θ$ and any arithmetical sentence $φ$, $Θ(\ulcornerφ\urcorner)\equiv φ$ 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.