Some results on the length of proofs

Abstract

Given a theory T, let ⊢ T k A \vdash _T^kA mean “A has a proof in T of at most k lines". We consider a formulation P A ∗ P{A^\ast } of Peano arithmetic with full induction but addition and multiplication being ternary relations. We show that ⊢ k A { \vdash ^k}A is decidable for P A ∗ P{A^\ast } and hence P A ∗ P{A^\ast } is closed under a weak ω \omega -rule. An analogue of Gödel’s theorem on the length of proofs is an easy corollary.