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 existe k ∈ ω s. t. $PA_M $ ┤ ∀x > k̄ A(x). The results imply that $PA_M $ does not prove scheme of induction or identity schemes in a bounded number of steps