Primitive Recursive Functions

1. Definition of recursive functions. In this paper, we shall consider certain reductions in the recursion scheme for defining primitive recursive functions. Hereafter, we shall refer to such functions simply as recursive functions. In §1, we define what is meant by a recursive function, and also define some recursive functions which will be used. The statement of the principal results of the paper will be found in §2. By a number, we shall mean one of the natural numbers 0, 1, 2, 3, • • • . We shall consider functions of any number of variables, each variable ranging over all numbers, and the values of the function being numbers. Small letters will denote variables assuming numerical values, and capital letters will denote functions. In the case of a function of one variable, we shall usually write Fx instead of F(x). A function will be called recursive if it can be obtained from certain initial functions by repeated substitution and recursion. The initial functions are the following: The identity functions ; that is, for every n and k with 1 ̂ k gw, the function Ink defined by