Models

A model for a list of axioms is an example of a system in which the axioms hold. It is the models in terms of which a list of axioms can be interpreted which make the axioms interesting.

But models can also be useful in examining the structure of systems of axioms. A famous example relates to Euclid's parallel postulate. Many mathematicians tried for a very long time to deduce this postulate from the other axioms of Euclidean geometry. All attempts were unsuccessful because the project is impossible. Models can be constructed in which all the other axioms of Euclidean geometry are true but the parallel postulate is false.

In this chapter, we construct a model for the real number axioms. The primary reason for producing this model is that its existence demonstrates that the axioms given for the real number system are consistent – i.e. they do not lead to a contradiction. It need hardly be said that an inconsistency in the axioms for the real number system would be a total disaster for the whole of analysis and most other branches of mathematics as well.

We have already seen (§9.21) that there is essentially only one model of the real number system – i.e. all models have precisely the same structure and differ only in the symbols which are used to label their elements. It therefore makes sense to identify a real number with the object we shall construct to represent it – i.e. to say that our model is the real number system.

This book contains an informal but systematic account of the logical and algebraic foundations of mathematical analysis written at a fairly elementary level. The book is entirely self-contained but will be most useful to students who have already taken, or are in the process of taking, an introductory course in basic mathematical analysis. Such a course necessarily concentrates on the notion of convergence and the rudiments of the differential and integral calculus. Little time is therefore left for consideration of the foundations of the subject. But the foundational issues are too important to be neglected or to be left entirely in the hands of the algebraists (whose views on what is important do not always coincide with those of an analyst). In particular, a good grasp of the material treated in this book is essential as a basis for more advanced work in analysis. The fact remains, however, that a quart will not fit into a pint bottle and only so many topics can be covered in a given number of lectures. In my own lecture course I deal with this problem to some extent by encouraging students to read the more elementary material covered in this book for themselves, monitoring their progress through problem classes. This seems to work quite well and it is for this reason that substantial sections of the text have been written with a view to facilitating ‘self-study’, even though this leads to a certain amount of repetition and of discussion of topics which some readers will find very elementary. Readers are invited to skip rather briskly through these sections if at all possible.