Truth, disjunction, and induction

Abstract

By a well-known result of Kotlarski, Krajewski, and Lachlan (1981), first-order Peano arithmetic PA can be conservatively extended to the theory CT − [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 results motivates the general question of determining natural axioms concerning the truth predicate that can be added to CT − [PA] while maintaining conservativity over PA. Our main result shows that conservativity fails even for the extension of CT − [PA] obtained by the seemingly weak axiom of disjunctive correctness DC that asserts that the truth predicate commutes with disjunctions of arbitrary finite size. In particular, CT − [PA] + DC implies Con(PA).Our main result states that the theory CT − [PA] + DC coincides with the theory CT 0 [PA] obtained by adding ∆ 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). Fedor Pakhamov's research is supported in part by the Young Russian Mathematics award. 2010 Mathematical Subject Classification: 03F30.CT − [PA] (compositional truth over PA with induction only for the language L A of arithmetic) is conservative over PA, i.e., if an L A -sentence ϕ is provable in CT − [PA], then ϕ is already provable in PA. New proofs of this conservativity result were given by Visser and Enayat [6] 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.On the other hand, it is well-known [9, Thm. 8.39 and Cor. 8.40] that the consistency of PA (and much more) is readily provable in the stronger theory CT [PA], which is the result of strengthening CT − [PA] with the scheme of induction over natural numbers for all L A+T -formulae, where L A+T := L A ∪ {T(x)}. 1 Indeed, it is straightforward to demonstrate the consistency of PA within the subsystem CT 1 [PA] of CT[PA], where CT n [PA] is the subtheory of CT[PA] with the scheme of induction over natural numbers limited to L A+T -formulae that are at most of complexity Σ n [16, Thm. 2.8]. The discussion above leaves open whether CT 0 [PA] is conservative over PA. Kotlarski [11] established that CT 0 [PA] is a subtheory of CT − [PA] + Ref(PA), where Ref(PA) is the L A+T -sentence stating that "every first order consequence of PA is true". Recently Le lyk [15] demonstrated that the converse also holds, which immediately implies that CT 0 [PA] is not conservative over PA since Con(P...

Hyperarithmetical Worm Battles

“…For languages extending the language of arithmetic, we require a way to finitely axiomatize Δ L 0 -induction, which is given for finite L (Lemma 4.2 in [7]). Over EA we have the following theorem.…”

Arithmetical and Hyperarithmetical Worm Battles

“…For languages extending the language of arithmetic, we require a way to finitely axiomatize ∆ L 0induction, which is given for finite L (Lemma 4.2 in [7]). Over EA we have the following theorem.…”

Disjunctions with stopping condition

“…Of course, the obstacle outlined above does not prove that our attempt will fail, but one can prove that it does using a recent result from [EP18]. It is shown there that CT − enriched with the principle: "a finite disjunction is true iff one of the disjuncts is true" is not conservative over PA and in fact has the same arithmetical strength as CT 0 , a compositional truth theory CT − with ∆ 0 induction for the formulae in the extended language.…”