Things that can and things that cannot be done in PRA

Abstract

It is well known by now that large parts of (non-constructive) mathematical reasoning can be carried out in systems $\text{T}$ which are conservative over primitive recursive arithmetic PRA (and even much weaker systems). On the other hand there are principles S of elementary analysis (like the Bolzano–Weierstraß principle, the existence of a limit superior for bounded sequences, etc.) which are known to be equivalent to arithmetical comprehension (relative to $\text{T}$) and therefore go far beyond the strength of PRA (when added to $\text{T}$). In this paper we determine precisely the arithmetical and computational strength (in terms of optimal conservation results and subrecursive characterizations of provably recursive functions) of weaker function parameter-free schematic versions $\text{S}{}^{\mathrm{-}}$ of S, thereby exhibiting different levels of strength between these principles as well as a sharp borderline between fragments of analysis which are still conservative over PRA and extensions which just go beyond the strength of PRA.