Things that can and things that cannot be done in PRA

It is well known by now that large parts of (non-constructive) mathematical reasoning can be carried out in systems 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 T) and therefore go far beyond the strength of PRA (when added to 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 S- 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. (C) 2000 Elsevier Science B.V. All rights reserved.