The Strength of Compositional Truth
Fangjing Xiong

Abstract:
A compositional theory of truth with the induction principle extended to sentences containing the truth predicate is not conservative over the base theory. It is unknown whether compositionality or extended induction contributes more to the nonconservativity result. This thesis follows Heck’s clarification by studying the strength of compositional truth, in particular, we study the strength of compositional truth in an induction-free environment.
We establish two conservation results concerning compositional truth without extended induction (whose axioms are denoted as CT) in an induction-free fragment of Peano Arithmetic PA−. We denote PA− with the compositional truth axioms CT[PA− ]. First, generalizing a model-theoretic proof of Enayat and Visser for the conservativity of CT[PA], we show that CT[PA− ] is syntactically conservative over PA− . Second, by generalizing Kaye’s modification of Lachlan’s proof of Lachlan’s theorem to PA− , we establish that CT[PA−] is not semantically conservative over PA− . Built from a lemma by Mateusz Łełyk, we also show that every computably enumerable extension of PA− has a non-recursively saturated model. On the technical side, the landscape regarding conservativity and recursive saturation for PA− is very similar to that of PA. On the philosophical side, compositional truth is much stronger than extended induction - it alone is sufficient to enforce recursive saturation.