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The Number Systems: Foundations of Algebra and Analysis
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AMS Chelsea Publishing
1964; 418 pp; hardcover
Volume: 333
Reprint/Revision History:
reprinted 1989; first AMS printing 2003
ISBN-10: 0-8218-2915-7
ISBN-13: 978-0-8218-2915-8
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 non-constructive set-theoretical 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 semi-constructive viewpoints, as has been demonstrated in more advanced texts and research articles."

• The Logical Background: 1.1 Introduction; 1.2 Logic
• The Set-Theoretical 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