Things that can and things that can't be done in PRA

Ulrich Kohlenbach

September 1998


It is well-known by now that large parts of (non-constructive) mathematical reasoning can be carried out in systems tex2html_wrap_inline32 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 tex2html_wrap_inline32) and therefore go far beyond the strength of PRA (when added to tex2html_wrap_inline32).
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 Stex2html_wrap_inline38 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

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