# The Foundations of Mathematics 2nd edition

Book Preface

The world has moved on since the first edition of this book was written on typewriters in 1976. For a start, the default use of male pronouns is quite rightly frowned upon. Educationally, research has revealed new insights into how individuals learn to thinkmathematically as they build on their previous experience (see [3]).1 We have used these insights to add comments that encourage the reader to reflect on their own understanding, thereby making more sense of the subtleties of the formal definitions. We have also added an appendix on self-explanation (written by Lara Alcock, Mark Hodds, and Matthew Inglis of the Mathematics Education Centre, Loughborough University) which has been demonstrated to improve longterm performance in making sense of mathematical proof. We thank the authors for their permission to reproduce their advice in this text.

The second edition has much in common with the first, so that teachers familiar with the first edition will find that most of the original content and exercises remain. However, we have taken a significant step forward. The first edition introduced ideas of set theory, logic, and proof and used them to start with three simple axioms for the natural numbers to construct the real numbers as a complete ordered field. We generalised counting to consider infinite sets and introduced infinite cardinal numbers. But we did not generalise the ideas of measuring where units could be subdivided to give an ordered field.

In this edition we redress the balance by introducing a new part IV that retains the chapter on infinite cardinal numbers while adding a new chapter on how the real numbers as a complete ordered field can be extended to a larger ordered field.

This is part of a broader vision of formal mathematics in which certain theorems called structure theorems prove that formal structures have natural interpretations that may be interpreted using visual imagination and symbolicmanipulation. For instance, we already know that the formal concept of a complete ordered field may be represented visually as points on a number line or symbolically as infinite decimals to perform calculations.

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