Truth, disjunction, and induction

Ali Enayat,Fedor Pakhomov

doi:10.1007/s00153-018-0657-9

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).

Highlights

By a theorem of Krajewski, Kotlarski, and Lachlan [12], every countable recursively saturated model M of PA (Peano Arithmetic) carries a ‘full satisfaction class’, i.e., there is a subset S of the universe M of M that ‘decides’ the truth/falsity of each sentence of arithmetic in the sense of M—even sentences of nonstandard length—while obeying the usual recursive clauses guiding the behavior of a Tarskian satisfaction predicate
This remarkable theorem implies that theory CT−[PA] is conservative over PA, i.e., if an LA-sentence φ is provable in CT−[PA], φ is already provable in PA
New proofs of this conservativity result were given by Visser and Enayat [5] using basic model theoretic ideas, and by Leigh [14] using proof theoretic tools; these new proofs make it clear that in the Krajewski–Kotlarski–Lachlan theorem the theory PA can be replaced by any ‘base’ theory that supports a modicum of coding machinery for handling elementary syntax

Summary

Introduction

By a theorem of Krajewski, Kotlarski, and Lachlan [12], every countable recursively saturated model M of PA (Peano Arithmetic) carries a ‘full satisfaction class’, i.e., there is a subset S of the universe M of M that ‘decides’ the truth/falsity of each sentence of arithmetic in the sense of M—even sentences of nonstandard length—while obeying the usual recursive clauses guiding the behavior of a Tarskian satisfaction predicate This remarkable theorem implies that theory CT−[PA] (compositional truth over PA with induction only for the language LA of arithmetic) is conservative over PA, i.e., if an LA-sentence φ is provable in CT−[PA], φ is already provable in PA. In 2012 Enayat found a proof of CT0[PA] within CT−[I 0 + Exp] + DC + IC; his proof was only privately circulated, and later was presented in the doctoral dissertation of Łełyk [15] This proof forms the content of Sect. In light of these developments, and the well-known conservativity of CT−[PA] + IC over PA (see Theorem 2.3), the question of conservativity of CT−[PA] + DC over PA came to prominence amongst truth theory experts [4, p.226], and had been unsuccessfully attacked by a number of researchers since 2013, until Pakhomov established IC within CT−[I 0 + Exp] + DC as in Sect. 3 of this paper, which, coupled with Enayat’s aforementioned result, yields Theorem 1 and exhibits the unexpected arithmetical strength of DC

Preliminaries

Disjunctive correctness implies inductive correctness

Closing remarks and open questions