On axiom schemes for T-provably Δ1 formulas

This paper investigates the status of the fragments of Peano Arithmetic obtained by restricting induction, collection and least number axiom schemes to formulas which are Δ1 provably in an arithmetic theory T. In particular, we determine the provably total computable functions of this kind of theories. As an application, we obtain a reduction of the problem whether IΔ0+¬exp implies BΣ1 to a purely recursion-theoretic question.Ministerio de Ciencia e Innovación MTM2008–0643

On a problem of J. Paris

We give alternative proofs for results of T. Slaman and N. Thapen, concerning the problem whether or not | Δ1 implies B Σ1. Our proofs isolate structures that could disprove this implication