Appendix 1 - Mathematical background

Summary

In this appendix we collect together some useful mathematical background. Readers may prefer to read the main text and refer to this appendix only if they get stuck. We do not give much in the way of proofs and the style is terse and rather dull, so this is not a substitute for standard texts. For example, Forster (2003) discusses in detail almost all the topics here, as well as much relevant material in logic and computability and some more advanced topics in set theory.

Mathematical notation and terminology

We use ‘iff’ as a shorthand for ‘if and only if’ and ‘w.r.t.’ for ‘with respect to’. We write x | y, read ‘x divides y’, to mean that y is an integer multiple of x, e.g. 3 | 6, 1 | x and x | 0. We use the usual arithmetic operations (‘+’ etc.) on numbers; we generally write xy for the product of x and y, but sometimes write x y to emphasize that there is an operation involved and make the syntax more regular. An operation such as addition for which the order of the two arguments is irrelevant (x + y = y + x) is called commutative, and an operation where the association does not matter (x + (y + z) = (x + y) + z) is said to be associative. We also use the conventional equality and inequality relations (‘=’, ‘≤’ etc.) on numbers, and sometimes emphasize that an equation is the definition of a concept by decorating the equality sign with def, e.g. tan(x) =def sin(x)/cos(x).