Introduction

In this chapter, a sequence of pure terms will be chosen to represent the natural numbers. It is then reasonable to expect that some of the other terms will represent functions of natural numbers, in some sense. This sense will be defined precisely below. The functions so representable will turn out to be exactly those computable by Turing machines.

In the 1930s, three concepts of computability arose independently: ‘Turing-computable function’, ‘recursive function’ and ‘λ-definable function’. The inventors of these three concepts soon discovered that all three gave the same set of functions. Most logicians took this as strong evidence that the informal notion of ‘computable function’ had been captured exactly by these three formally-defined concepts.

Here we shall look at the recursive functions, and prove that all these functions can be represented in λ and CL. (We shall not work with the Turing-computable functions because their representability-proof is longer.)

An outline definition of the recursive functions will be given here; more details and background can be found in many textbooks on computability or textbooks on logic which include computability, for example [Coh87], [Men97] or the old but thorough [Kle52].

Notation 4.1 This chapter is written in the same neutral notation as the last one, and its results will hold for both λ and CL unless explicitly stated otherwise.