Truth, disjunction, and induction

Abstract

By a well-known result of Kotlarski et al. (1981), first-order Peano arithmetic \({{\mathsf {P}}}{{\mathsf {A}}}\) can be conservatively extended to the theory \({{\mathsf {C}}}{{\mathsf {T}}}^{-}\mathsf {[PA]}\) of a truth predicate satisfying compositional axioms, i.e., axioms stating that the truth predicate is correct on atomic formulae and commutes with all the propositional connectives and quantifiers. This result motivates the general question of determining natural axioms concerning the truth predicate that can be added to \({{\mathsf {C}}}{{\mathsf {T}}}^{-}\mathsf {[PA]}\) while maintaining conservativity over \( {{\mathsf {P}}}{{\mathsf {A}}}\). Our main result shows that conservativity fails even for the extension of \({{\mathsf {C}}}{{\mathsf {T}}}^{-}\mathsf {[PA]}\) obtained by the seemingly weak axiom of disjunctive correctness \({{\mathsf {D}}}{{\mathsf {C}}}\) that asserts that the truth predicate commutes with disjunctions of arbitrary finite size. In particular, \({{\mathsf {C}}}{\mathsf {T}}^{-}\mathsf {[PA]}+\mathsf {DC}\) implies \(\mathsf {Con}(\mathsf {PA})\). Our main result states that the theory \({\mathsf {C}}{\mathsf {T}}^{-}\mathsf {[PA]}+\mathsf {DC}\) coincides with the theory \({\mathsf {C}}{\mathsf {T}}_{0}\mathsf {[PA]}\) obtained by adding \( \Delta _{0}\)-induction in the language with the truth predicate. This result strengthens earlier work by Kotlarski (1986) and Cieśliński (2010). For our proof we develop a new general form of Visser’s theorem on non-existence of infinite descending chains of truth definitions and prove it by reduction to (Löb’s version of) Gödel’s second incompleteness theorem, rather than by using the Visser–Yablo paradox, as in Visser’s original proof (1989).