AMS Chelsea Publishing 1964; 418 pp; hardcover Volume: 333 Reprint/Revision History: reprinted 1989; first AMS printing 2003 ISBN10: 0821829157 ISBN13: 9780821829158 List Price: US$54 Member Price: US$48.60 Order Code: CHEL/333.H
 The subject of this book is the successive construction and development of the basic number systems of mathematics: positive integers, integers, rational numbers, real numbers, and complex numbers. This second edition expands upon the list of suggestions for further reading in Appendix III. From the Preface: "The present book basically takes for granted the nonconstructive settheoretical foundation of mathematics, which is tacitly if not explicitly accepted by most working mathematicians but which I have since come to reject. Still, whatever one's foundational views, students must be trained in this approach in order to understand modern mathematics. Moreover, most of the material of the present book can be modified so as to be acceptable under alternative constructive and semiconstructive viewpoints, as has been demonstrated in more advanced texts and research articles." Table of Contents  The Logical Background: 1.1 Introduction; 1.2 Logic
 The SetTheoretical Background: 2.1 Sets; 2.2 An algebra of sets; 2.3 Relations and functions; 2.4 Mathematical systems of relations and functions
 The Positive Integers: 3.1 Basic properties; 3.2 The arithmetic of positive integers; 3.3 Order; 3.4 Sequences, sums and products
 The Integers and Integral Domains: 4.1 Toward extending the positive integers; 4.2 Integral domains; 4.3 Construction and characterization of the integers; 4.4 The integers as an indexing system; 4.5 Mathematical properties of the integers; 4.6 Congruence relations in the integers
 Polynomials: 5.1 Polynomial functions and polynomial forms; 5.2 Polynomials in several variables
 The Rational Numbers and Fields: 6.1 Toward extending integral domains; 6.2 Fields of quotients; 6.3 Solutions of algebraic equations in fields; 6.4 Polynomials over a field
 The Real Numbers: 7.1 Toward extending the rationals; 7.2 Continuously ordered fields; 7.3 Infinite series and representations of real numbers; 7.4 Polynomials and continuous functions on the real numbers; 7.5 Algebraic and transcendental numbers
 The Complex Numbers: 8.1 Basic properties; 8.2 Polynomials and continuous functions in the complex numbers; 8.3 Roots of complex polynomials
 Algebraic Number Fields and Field Extensions: 9.1 Generation of subfields; 9.2 Algebraic extensions; 9.3 Applications to geometric construction problems; 9.4 Conclusion
 Appendix I: Some axioms for set theory
 Appendix II: The analytical basis of the trigonometric functions
 Bibliography
 Index
