A Formal System of Partial Recursive Functions

Abstract

We know that many parts of ordinary recursive function theory can be developed formally in a certain extension of the formal number theory (e.g. Peano arithmetic). But we encounter some difficulties when we want to deal with partial recursive functions, since in ordinary logical calculi only total functions and predicates can be treated. The most natural way to treat partial functions will be to take their graphs instead of functions themselves. More precisely, to represent an «-ary partial functions, an (rc + l)-ary predicate P having the property that P(xl5..., xn9 y) holds for at most one y for any xl9...,xn is taken. However, it will entail considerable complications to express properties of partial recursive functions in such forms. In this paper, we shall attempt to formalize the theory of partial recursive functions, which is called PRN, on a logical calculus in which partial functions and predicates can be treated. For the logical calculus mentioned above, we shall take a system which is obtained from the one introduced by Ebbinghaus [1] by extending it to second order. We shall introduce some extensions of PRN and examine their logical powers. Our approach contrasts with the one by Scott [8]. In [8] a partial function from a set A to another set B is regarded as a total function from A to the set B with the element which represents the undefined value. In § 1 and § 2, the second order logic of partial functions SP and its semantics are introduced. Axioms of the theory PRN are given in §3. In §4, some completeness results for some extensions of PRN are proved and applications of our theories to the mathematical theory of computation are suggested.