Friedberg numbering in fragments of Peano Arithmetic and α-recursion theory

In this paper, we investigate the existence of a Friedberg numbering in fragments of Peano Arithmetic and initial segments of Gödel’s constructible hierarchy Lα, where α is Σ1 admissible. We prove that (1) Over P− +BΣ2, the existence of a Friedberg numbering is equivalent to IΣ2, and (2) For Lα, there is a Friedberg numbering if and only if the tame Σ2 projectum of α equals the Σ2 cofinality of α.