Conservativity for Theories of Compositional Truth via Cut Elimination

Abstract

We present a cut elimination argument that witnesses the conservativity of the compositional axioms for truth (without the extended induction axiom) over any theory interpreting a weak subsystem of arithmetic. In doing so we also fix a critical error in Halbach's original presentation. Our methods show that the admission of these axioms determines a hyper-exponential reduction in the size of derivations of truth-free statements. §1. Overview. Let IΔ 0 + exp and IΔ 0 + exp 1 be the first-order theories extending Robinson's arithmetic by Δ 0 -induction and, respectively, axioms expressing the totality of the exponentiation and hyper-exponentiation function. If S is a first-order theory interpreting IΔ 0 + exp then by CT[S] we denote the extension of S by a fresh unary predicate T and the compositional axioms of truth for T. 1 In this paper we provide syntactic proofs for the following theorems.Theorem 1.1. Let S be an elementary axiomatised theory in a finite language interpreting IΔ 0 + exp. Then CT[S] conservatively extends S. Moreover, this fact is verifiable in IΔ 0 + exp 1 .Let p be a fresh unary predicate symbol not present in the language L of S. An L-formula D is an S-schema if S D → for every L-formula and there exists a finite set of formulae U such that S Dx → ∃ ϕ∈U (x = ϕ[ /p] ). Theorem 1.2. Let S be as above. For any S-schema D, the theory CT[S] + ∀x(Dx → Tx) is a conservative extension of S. Moreover, this fact is verifiable in IΔ 0 + exp 1 .The first part of Theorem 1.1 was first established by Barwise and Schlipf in the early 70s (see Theorem IV.5.3 of [1]) and later independently proved by Kotlarski, Krajewski, and Lachlan [13] for the case of S = PA, also establishing the first part of Theorem 1.2 in this case. Both proofs are model-theoretic, showing that a countable nonstandard model of S contains a full satisfaction class if it is recursively saturated. Since every model of S is elementarily extended by a recursively saturated model of the same cardinality, conservativity is obtained. Recently, Enayat and

Truth and Feasible Reducibility

“…Our proof of the feasible reduction of CT − [PA] to PA includes the verification that PA proves the formal consistency of every finite subtheory of CT − [PA], thereby establishing that CT − [PA] is a reflexive theory. This result follows from Leigh's work [15]; and was also established by Enayat and Visser (unpublished) with help of the "low basis theorem" of computability theory to arithmetize their model-theoretic proof of conservativity of CT − [PA] over PA. The proof presented here, however, is based on a simpler arithmetization of the Enayat-Visser construction and does not appeal to the low basis theorem; the syntactic analysis of this arithmetization forms one of the main ingredients of the proof of our main result.…”

“…2 This equivalence follows from two key facts: (1) every countable consistent theory has a countable recursively saturated model, and (2) countable recursively saturated models are resplendent, both of which can be verified in the subsystem ACA0 of second order arithmetic.3 Supexp asserts the totality of the superexponential function Supexp(n, x), with Supexp(0, x) = x and Supexp(n + 1, x) = 2 Supexp(n,x) . Leigh [15] refers to this function as hyper-exponentiaton. 4 The choice of the "standard proof system" is immaterial since it is well-known that any two such systems polynomially simulate each other [18].…”

“…3 Supexp asserts the totality of the superexponential function Supexp(n, x), with Supexp(0, x) = x and Supexp(n + 1, x) = 2 Supexp(n,x) . Leigh [15] refers to this function as hyper-exponentiaton. 4 The choice of the "standard proof system" is immaterial since it is well-known that any two such systems polynomially simulate each other [18].…”

“…over PA ( [13], [15]) but not feasibly reducible to PA since it has superexponential speed-up over PA [7].…”

Unifying the Philosophy of Truth

“…Happily, Leigh (2012) has succeeded in developing a proof-theoretic demonstration of the conservativity of PA FT over PA that is implementable in PRA. Happily, Leigh (2012) has succeeded in developing a proof-theoretic demonstration of the conservativity of PA FT over PA that is implementable in PRA.…”