The interpretability logic of Peano arithmetic

Abstract

PA is Peano arithmetic. The formula InterpPA(α, β) is a formalization of the assertion that the theory PA + α interprets the theory PA + β (the variables α and β are intended to range over codes of sentences of PA). We extend Solovay's modal analysis of the formalized provability predicate of PA, PrPA(x), to the case of the formalized interpretability relation InterpPA(x, y). The relevant modal logic, in addition to the usual provability operator ‘□’, has a binary operator ‘⊳’ to be interpreted as the formalized interpretability relation. We give an axiomatization and a decision procedure for the class of those modal formulas that express valid interpretability principles (for every assignment of the atomic modal formulas to sentences of PA). Our results continue to hold if we replace the base theory PA with Zermelo-Fraenkel set theory, but not with Gödel-Bernays set theory. This sensitivity to the base theory shows that the language is quite expressive. Our proof uses in an essential way earlier work done by A. Visser, D. de Jongh, and F. Veltman on this problem.

Interpretability suprema in Peano Arithmetic

“…Švejdar's construction of suprema in (V , ) (see (1), or Theorem 5 in Sect. 4) can be seen as a way of compressing the information in S into a single sentence.…”

“…We have x+1 x+1 A by GL for IΣ x , and x+1 ¬ Π 1 x ⊥ by reflection. Therefore x+1 ( x+1 A ∧ ¬ Π 1 x ⊥), and thus clearly also (22).…”

“…It was proven independently by Berarducci [1] and Shavrukov [24] that the theorems of ILM are exactly the propositional schemata involving formalized provability and interpretability that are provable in PA.…”

Problems in the Logic of Provability

“…16 Specifically, we want S to be an extension of Buss's theory S 1 2 (see [24] or [48]). We assume that the axioms of T are given by a ∆ b 1 -formula.…”

Unary interpretability logic.

“…On the modal side the reduction of unary to binary interpretability leads to the following definition: Definition 1. 1 Define in £ (D, >) the unary interpretability operator T by IA := T > A, and let <£(Π,I) extend £(B) with I.…”