Parameter Free Induction and Reflection

We give a precise characterization of parameter free n and n induction schemata I n and I n in terms of re ection principles This allows us to show that I n is conservative over I n w r t boolean combinations of n sentences for n In particular we give a positive answer to a question by R Kaye whether the provably recursive functions of I are exactly the primitive recursive ones We also obtain sharp results on the strength of bounded number of instances of parameter free induction in terms of iterated re ection