Foundations of Mathematical Analysis

Foundations of Mathematical Analysis covers a wide variety of topics that will be of great interest to students of pure mathematics or mathematics and philosophy. Aimed principally at postgraduates and well-motivated undergraduates, its primary concern is a discussion of the fundamental number systems, S \Bbb NS , S \Bbb ZS , S \Bbb QS , S \Bbb RS , and S \Bbb CS , in the context of the branches of mathematics for which they form a starting point; for example, a study of the natural numbers leads on to logic (via G\"odel's theorems), and of the real numbers (as constructed by Cauchy) to metric spaces and topology. Prof. Truss offers a refreshingly original approach to these matters, presenting standard material in new ways, and incorporating less mainstream topics such as long real and rational lines and the p-adic numbers. With a discussion of constructivism and independence questions including Suslin's problem and the continuum hypothesis, Prof. Truss completes a wide-ranging consideration of the development of mathematics from the very beginning, concentrating on the foundational issues particularly related to analysis. The book is presented in such a manner as to be accessible to non-specialists.

Table of Contents:
1. The natural numbers; 2. Some set theory; 3. The integers; 4. The rational numbers; 5. The real numbers; 6. Metric spaces; 7. Beginnings of analysis; 8. The complex numbers; 9. Irrational numbers; 10. Classical spaces associated with R; 11. Measure and category; 12. The continuum problem; 13. Constructive analysis; References; Index

Review :
"The book can be warmly recommended to students interested to find various connections and intersection of analysis and other braches of classical mathematics and provides a very interesting second reading for virtually everyone."--European Mathematical Society Newsletter "This book is a remarkable attempt to describe the foundations of mathematical analysis, starting with the logical development and ending with Lebesgue theory and the topology of the real line. . . . [M]any less familiar topics are included and chapters with familiar-sounding titles appear to be treated in novel ways. For instance, the first chapter about natural numbers contains a very well written sketch about coding and the Gödel theorems. . . . The chapters about the integers and the rationals are written with emphasis on their algebraic properties. This makes it possible for the author to sketch elementary Galois theory and the proof of the fundamental theorem of algebra in his chapter about complex numbers. The three chapters about real numbers, metric spaces and the beginnings of analysis contain many interesting topics . . . The book is written in a brilliant style, perfectly readable for persons with some experience in mathematics."--Mathematical Reviews