Realizability Interpretation of PA by Iterated Limiting PCA

Abstract

For any partial combinatory algebra (PCA for short) A, the class of A-representable partial functions from N to A quotiented by the filter of cofinite sets of N, is a PCA such that the representable partial functions are exactly the limiting partial functions of A-representable partial functions(Akama, "Limiting partial combinatory algebras" Theoret. Comput. Sci. Vol.311 2004). The n-times iteration of this construction results in a PCA that represents any n-iterated limiting partial recursive functions, and the inductive limit of the PCAs over all n is a PCA that represents any arithmetical, partial function. Kleene's realizability interpretation over the former PCA interprets the logical principles of double negation elimination for \Sigma^0_n-formulas, and that over the latter PCA interprets Peano's arithmetic (PA for short). A hierarchy of logical systems between Heyting's arithmetic and PA is used to discuss the prenex normal form theorem, the relativized independence-of-premise schemes, and "PA is an unbounded extension of HA."