Restricted polynomial induction versus parameter free ordinary induction
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 $I\Delta _0$ and, on the other hand, extensions 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 $I\Delta _0$ in which the totality of the operations is not assumed.