The foundations of mathematics

Part I. The intuitive background

1. Mathematical thinking

2. Number systems

Part II. The beginnings of formalisation

3. Sets

4. Relations

5. Functions

6. Mathematical logic

7. Mathematical proof

Part III. The development of axiomatic systems

8. Natural numbers and proof by induction

9. Real numbers

10. Real numbers as a complete ordered field

11. Complex numbers and beyond

Part IV. Using axiomatic systems

12. Axiomatic systems, structure theorems, and flexible thinking

13. Permutations and groups

14. Cardinal numbers

15. Infinitesimals

Part V. Strengthening the foundations

16. Axioms for set theory.

Cover; Preface to the Second Edition; Preface to the First Edition; Contents; Part I The Intuitive Background; 1 Mathematical Thinking; 2 Number Systems; Part II The Beginnings of Formalisation; 3 Sets; 4 Relations; 5 Functions; 6 Mathematical Logic; 7 Mathematical Proof; Part III The Development of Axiomatic Systems; 8 Natural Numbers and Proof by Induction; 9 Real Numbers; 10 Real Numbers as a Complete Ordered Field; 11 Complex Numbers and Beyond; Part IV Using Axiomatic Systems; 12 Axiomatic Systems, Structure Theorems, and Flexible Thinking; 13 Permutations and Groups; 14 Cardinal Numbers.

15 InfinitesimalsPart V Strengthening the Foundations; 16 Axioms for Set Theory; Appendix-How to Read Proofs: The 'Self-Explanation' Strategy; References and Further Reading; Index.