Fragment of Nonstandard Analysis with a Finitary Consistency Proof

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∆0 + EXP, and on the other hand, NQA+ interprets the theories IΣ1 and WKL0. §