Restricted polynomial induction versus parameter free ordinary induction

Abstract

. The paper is a continuation of [Z. Adamowicz, Fund. Math. 242 (2018)]. We consider conservativity questions between, on the one hand, arithmetical theories in which the operations of successor, addition and multiplication are not provably total and which are fragments of the bounded arithmetic theory I ∆ 0 and, on the other hand, exten-sions of those theories to subtheories of Buss’s bounded arithmetic S 2 . These questions are related to the problem of finite axiomatizability of a version of I ∆ 0 in which the totality of the operations is not assumed.