Parameter Free Induction and Provably Total Computable Functions

We study the classes of computable functions that can be proved to be total by means of parameter free C, and 4 induction schemata, ZC; and ZlI;, over Kalmar elementary arithmetic. We give a positive answer to a question, whether the provably total computable functions of Zq are exactly the primitive recursive ones, and show that the class of such functions for ICI + I% coincides with the class of doubly recursive functions of Peter. We also characterize provably total computable functions of theories of the form ZIInQ, and IC, + ZII,$ for all n > 1, in terms of the fast growing hierarchy. These results are based on a precise characterization of ZC; and Zw in terms of reflection principles and conservation results for local reflection principles obtained by techniques of modal provability logic. Using similar ideas we show that ZII”;, is conservative over ZC; w.r.t. boolean combinations of &+I sentences, for n 2 1, and obtain a number of results on the strength of bounded number of instances of parameter free induction schemata and complexity of their axiomatization. @ 1999 Elsevier Science B.V. All rights reserved.