Strictly Primitive Recursive Realizability, II. Completeness with Respect to Iterated Reflection and a Primitive Recursive $\omega$-Rule

Abstract

The notion of strictly primitive recursive realizability is further investigated, and the realizable prenex sentences, which coincide with primitive recursive truths of classical arithmetic, are characterized as precisely those provable in transfinite progressions $\{\mathrm{PRA}(b) \vert b \in \underline{\mathrm{O}}\}$ over a fragment $\mbox{PR-}(\Sigma^{0}_{1}\mbox{-IR})$ of intuitionistic arithmetic. The progressions are based on uniform reflection principles of bounded complexity iterated along initial segments of a primitive recursively formulated system $\mathrm{\underline{O}}$ of notations for constructive ordinals. A semiformal system closed under a primitive recursively restricted $\omega$-rule is described and proved equivalent to the transfinite progressions with respect to the prenex sentences.