Chapter 11 Appendix B - Peano’s Axioms

At some point in your life I would imagine you have asked yourself the question ‘What exactly are numbers?’. You weren’t the first! Many of us have encountered a similar question but answering it is not quite so straightforward. We often hear the response ‘Numbers are an abstract concept’, and while there may be some merit in this approach, it doesn’t really provide much of an answer, and I would imagine that you are left feeling a bit ‘short changed’ by this reply. In this short report, we start by considering natural numbers and we shall look at our question from two points of view. First we look at the basic properties we (expect) natural numbers to have. Can we develop a (short) collection of axioms, or basic truths, that the natural numbers should satisfy and convince ourselves that, effectively, all other properties of the natural numbers can be derived from these axioms? Next we consider whether there is a set of (mathematical) objects that possess these properties. If so then perhaps this set may be taken as a model for the natural numbers. Finally we briefly consider how to extend our work on natural numbers to ‘define’ the integers and then the rationals. The real numbers must wait for another day!

Giuseppe Peano was born in Italy in 1858, the son of a farmer. He was taken to Turin by his uncle in 1870 and entered the University of Turin in 1876. He published his famous axioms (or postulates as they are sometimes know) in 1889, curiously in Latin, in a pamphlet entitled Arithmetices principia, nova methodo exposita. A number of other mathematicians, among them Richard Dedekind, had been working on similar axioms at about the same time. Dedekind wrote a paper in 1888 entitled ‘Was sind und was sollen die Zahlen?’ (What are and what should the numbers be?), and the Peano axioms are sometimes referred to as the Dedekind-Peano axioms.

Peano’s axioms define the arithmetical properties of the natural numbers and provide us with a logical foundation on which to build a study of their more advanced properties.

You should be aware that most of the material in this short collection of notes is actually beyond the scope of our 1st year introductory course in Number Theory and is included only for the benefit of those who are interested in the foundations of numbers. You should not worry unduly if you find this material hard! There will be NO exam on it! It should also be noted that a full treatment of this topic would require much more rigour and depth than we can give it at this stage, so you should treat these notes as an outline of the topic. Nevertheless, I hope there is enough detail to spark your interest in some of the more subtle areas of Pure Mathematics, and encourage you to further study.