Infinitesimal numbers

The Completeness and Compactness Theorems for first-order logic are interesting from the point of view of the foundations of mathematics, which is what they were originally designed for, but they also provide a powerful logical toolkit that can be applied to other areas of mathematics. One of the most exciting applications of the Completeness and Compactness Theorems is the discovery by Robinson that they may be used to make perfectly rigorous sense of the idea of an infinitesimal number, and to use infinitesimals to present the material of traditional analysis, including continuity and differentiability. Robinson called his method ‘nonstandard analysis’, which to my mind is a somewhat unfortunate name as there is nothing at all improper about his approach. Indeed, if historical circumstances had been different, nonstandard analysis might even have been mainstream analysis. That it is not is possibly due to the logical difficulties some mathematicians have in understanding how the analysis is set up – difficulties we aim to set to rights in this chapter.

Throughout this chapter I shall spell the word ‘nonstandard’ without a hyphen, to emphasise that this word is being used in the technical sense of ‘pertaining to infinite or infinitesimal numbers’, and not in the more common everyday sense of ‘not standard’ – which will never be used and always spelled ‘non-standard’.

The nonstandard method involves using methods from logic to build an extended version of the real number line with infinitesimals (called the ‘hyperreal number line’) and moving between the hyperreals and the usual reals. There are several possible approaches to this, including axiomatic ways that make the job of transferring information between the hyperreals and the reals almost completely automatic.