The optimality of induction as an axiomatization of arithmetic

Abstract

By induction for a formula φ we mean the schema(where the terms in brackets are implicitly substituted for some fixed variable, with the usual restrictions). Let be the schema IAφ for φ in Πn (i.e. ); similarly for . Each instance of is Δn+2, and each instance of is Σn+1 Thus the universal closure of an instance α is Πn+2 in either case. Charles Parsons [72] proved that and are equivalent over Z0, where Z0 is essentially Primitive Recursive Arithmetic augmented by classical First Order Logic [Parsons 70].Theorem. For each n > 0 there is a Πn formula π for whichis not derivable in Z0from(i) true Πn+1sentences; nor even(ii) Πn+1sentences consistent withZ0.