Abstract

We study the classes of computable functions that can be proved to be total by means of parameter free Σ_n and Π_n, induction schemata, IΣ_n⁻ and IΠ_n⁻, over Kalmar elementary arithmetic. We give a positive answer to a question, whether the provably total computable functions of IΠ₂⁻ are exactly the primitive recursive ones, and show that the class of such functions for IΣ₁ + IΠ₂⁻ coincides with the class of doubly recursive functions of Peter. We also characterize provably total computable functions of theories of the form IΠ_{n + 1}⁻ and IΣ_n + IΠ_{n + 1}⁻ for all n 1, in terms of the fast growing hierarchy.

These results are based on a precise characterization of IΣ_n⁻ and IΠ_n⁻ 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 IΠ_{n + 1}⁻ is conservative over IΣ_n⁻ w.r.t. boolean combinations of Σ_{n + 1} sentences, for n 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.

Author Keywords: Parameter free induction; Provably total computable functions; Reflection principles; Fast growing hierarchy

Mathematical subject codes: 03F30; 03D20

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¹ Supported by the Russian Foundation for Basic Research grant 96-01-01222.