2 - Functions and enumerations

Summary

We start by fixing some entirely standard notation and terminology for talking about functions (worth knowing anyway, quite apart from the occasional use we make of it in coming chapters). We next introduce the useful little notion of a ‘characteristic function’. Then we explain the idea of enumerability and give our first example of a ‘diagonalization argument’ – an absolutely crucial type of argument which will feature repeatedly in this book.

Kinds of function

(a) Functions, and in particular functions from natural numbers to natural numbers, will feature pivotally in everything that follows.

Note though that our concern will be with total functions. A total one-place function maps each and every element of its domain to some unique corresponding value in its codomain. Similarly for many-place functions: for example, the total two-place addition function maps any two numbers to their unique sum.

For certain wider mathematical purposes, especially in the broader theory of computation, the more general idea of a partial function can take centre stage. This is a mapping f which does not necessarily have an output for each argument in its domain (for a simple example, consider the function mapping a natural number to its natural number square root, if it has one). However, we won't need to say much about partial functions in this book, and hence – by default – plain ‘function’ will henceforth always mean ‘total function’.

(b) …