We introduce a nonstandard arithmetic $NQA^-$ based on the theory developed by R. Chuaqui and P. Suppes in [2] (we will denote it by $NQA^+$ ), with a weakened external open minimization schema. A finitary consistency proof for $NQA^-$ formalizable in PRA is presented. We also show interesting facts about the strength of the theories $NQA^-$ and $NQA^+$ ; $NQA^-$ is mutually interpretable with $I\Delta_0 + EXP$ , and on the other hand, $NQA^+$ interprets the theories IΣ1 and $WKL_0$.

It is well known by now that large parts of mathematical reasoning can be carried out in systems which are conservative over primitive recursive arithmetic PRA . On the other hand there are principles S of elementary analysis which are known to be equivalent to arithmetical comprehension and therefore go far beyond the strength of PRA . In this paper we determine precisely the arithmetical and computational strength 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. (shrink)