  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."
