2023/11/22 by Piotr Gruza, Gruza, Piotr, Mateusz Łełyk +1
Computer Science · Mathematics · #03A05 (Secondary) #03F30 (Primary) #03F40 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2311.13519
openalex publication_date 2023/11/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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)≡ 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 Σ2-definable in the standard model of arithmetic. We conclude by remarking that no Σ2-sentence, satisfying certain further natural conditions, can be a definition of truth for the language of arithmetic.