Kreisel's Conjecture with minimality principle

Abstract

We prove that Kreisel's Conjecture is true, if Peano arithmetic is axiomatised using minimality principle and axioms of identity (theory PA_M). The result is independent on the choice of language of PA_M. We also show that if infinitely many instances of A(x) are provable in a bounded number of steps in PA_M then there exists k∈ω s.t. PA_M⊢ ∀ x > \overline{k} A(x). The results imply that PA_M does not prove scheme of induction or identity schemes in a bounded number of steps.