We show that a typed compositional theory of positive truth with internal induction for total formulae (denoted by PTtot) is not semantically conservative over Peano arithmetic. In addition, we observe that the class of models of PA expandable to models of PTtot contains every recursively saturated model of arithmetic. Our results point to a gap in the philosophical project of describing the use of the truth predicate in model-theoretic contexts.

We introduce a tool for analysing models of $\text {CT}^-$ , the compositional truth theory over Peano Arithmetic. We present a new proof of Lachlan’s theorem that the arithmetical part of models of $\text {CT}^-$ are recursively saturated. We also use this tool to provide a new proof of theorem from [8] that all models of $\text {CT}^-$ carry a partial inductive truth predicate. Finally, we construct a partial truth predicate defined for a set of formulae whose syntactic depth forms a nonstandard cut which cannot be extended to a full truth predicate satisfying $\text {CT}^-$ .

We prove that the theory of the extensional compositional truth predicate for the language of arithmetic with ∆ 0 -induction scheme for the truth predicate and the full arithmetical induction scheme is not conservative over Peano Arithmetic. In addition, we show that a slightly modified theory of truth actually proves the global reflection principle over the base theory.

Let T be any of the three canonical truth theories CT − (Compositional truth without extra induction), FS − (Friedman-Sheard truth without extra induction), and KF − (Kripke-Feferman truth without extra induction), where the base theory of T is PA (Peano arithmetic). We show that T is feasibly reducible to PA, i.e., there is a polynomial time computable function f such that for any proof π of an arithmetical sentence φ in T , f (π) is a proof of φ in PA. In particular, T has at most polynomial speed-up over PA, in sharp contrast to the situation for T [B] for finitely axiomatizable base theories B. 5 Open Questions 46 6 Appendix 46 1 Introduction One of the celebrated results in the area of axiomatic theories of truth is the Krajewski-Kotlarski-Lachlan (KKL) theorem [13] that asserts that every countable recursively saturated model of PA (Peano arithmetic) is expandable to a model of CT − [PA] (compositional truth over PA with no extra induction 1 ). The KKL theorem is an overtly model-theoretic result, but it is well-known that it is equivalent to the conservativity of CT − [PA] over PA. 2 Recent proofs of the KKL theorem given by Enayat and Visser [5] (using model-theoretic techniques) and Leigh [15] (using proof-theoretic machinery) show that CT − [B] is conservative over B for every "base theory" B (i.e., a theory B that supports a modicum of coding machinery for handling elementary syntax). Leigh's proof makes it clear that if B is a base theory with a computable set of axioms, then CT − [B] is proof-theoretically reducible to B, and in particular, there is a primitive recursive function f such that for any proof π of a sentence φ in CT − [B], where φ is a sentence in the language of B, f (π) is a proof of φ in B. Indeed, Leigh's "reducing function" f is readily seen to be a provably total function of the fragment of PRA (Primitive Recursive Arithmetic) commonly known asThe main result of this paper shows that CT − [PA] is feasibly reducible to PA, i.e., there is a polynomial-time computable function f such that for any proof π of an arithmetical sentence φ in CT − [PA], f (π) is a proof of φ in PA. The feasible reducibility of CT − [PA] to PA readily implies that CT − [PA] does not exhibit significant speed-up over PA, i.e., there is a polynomial function p(x) such that for any arithmetical sentence φ, if φ is provable in CT − [PA] by a proof of length n (in some standard proof system 4 ), then φ is provable in PA by a proof of length p(n). This solves a problem posed by Enayat in 2012 [4]. The absence of significant speed up of CT − [PA] over PA implied by the feasible reducibility of CT − [PA] to PA exhibits a dramatic difference between CT − [PA] and CT − [B] for finitely axiomatized base theories B, since as shown by Fischer [7], CT − [B] has superexponential speed-up over B for finitely axiomatized base theories B, and therefore, CT − [B] is not feasibly reducible to B for finitely axiomatized base theories B. It is also known that CT − [PA] + Int (where Int is the axiom of internal induction) is conse...