Foundations of Constructive Mathematics Metamathematical Studies
This book is about some recent work in a subject usually considered part of "logic" and the" foundations of mathematics", but also having close connec tions with philosophy and computer science. Namely, the creation and study of "formal systems for const
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Foundations of Constructive Mathematics Metamathematical Studies
Springer-Verlag Berlin Heidelberg NewYork Tokyo
Michael J. Beeson Department of Mathematics and Computer Science San Jose State University San Jose, CA 95192, USA
AMS Subject Classification (1980): 03F50, 03F55, 03F60, 03F65 ISBN-13: 978-3-642-68954-3 DOl: 10.1007/978-3-642-68952-9
e-ISBN-13: 978-3-642-68952-9
Library of Congress Cataloging in Publication Data Beeson, Michael J., 1945-. Foundations of constructive mathematics. (Ergebnisse der Mathematik und ihrer Grenzgebiete; 3. Folge, Bd. 6) Bibliography: p. Includes indexes. I. Constructive mathematics. 1. Title. II. Series. QA9.56.B44 1985 511.3 84-10583 ISBN 0-387-12173-0 (U.S.) This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illus· trations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. © by Springer-Verlag Berlin Heidelberg 1985 Softcover reprint of the hardcover 1st edition 1985 Typesetting, printing and binding: Universitiitsdruckerei H. Stiirtz AG, 8700 Wiirzburg 2141/3140-543210
Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge . Band 6 A Series of Modem Surveys in Mathematics
Editorial Board E. Bombieri, Princeton S. Feferman, Stanford N.H. Kuiper, Bures-sur-Yvette P. Lax, NewYork R Remmert (Managing Editor), Miinster W. Schmid, Cambridge, Mass. J-P. Serre, Paris 1. Tits, Paris
This book is dedicated to the memories of Errett Bishop and Arend Heyting
Table of Contents
Introduction User's Manual Common Notations Acknowledgements
XIII XVIII XX XXII
Part One. Practice and Philosophy of Constructive Mathematics
1
Chapter I. Examples of Constructive Mathematics
3
1. The Real Numbers 2. Constructive Reasoning . . . . . 3. Order in the Reals . . . . . . . 4. Sub fields of R with Decidable Order 5. Functions from Reals to Reals 6. Theorem of the Maximum . 7. Intermediate Value Theorem 8. Sets and Metric Spaces 9. Compactness ..... . 10. Ordinary Differential Equations 11. Potential Theory . 12. The Wave Equation . 13. Measure Theory 14. Calculus of Variations 15. Plateau's Problem 16. Rings, Groups, and Fields 17. Linear Algebra . . . . 18. Approximation Theory 19. Algebraic Topology . . 20. Standard Representations of Metric Spaces 21. Some Assorted Problems ...... .
3 5 6 8 9 10 11 12 13 14 15 17 17 19 20 22 24 25
Chapter II. Informal Foundations of Constructive Mathematics
33
1. Numbers 2. Operations or Rules 3. Sets and Presets
26
27 30
33
34 34
VIII
4. 5. 6. 7. 8. 9. 10.
Table of Contents
Constructive Proofs Witnesses and Evidence Logic . . . . . Functions . . . . . . Axioms of Choice Ways of Constructing Sets Definite Presets ....
Chapter III. Some Different Philosophies of Constructive Mathematics 1. 2. 3. 4. 5. 6. 7. 8.
The Russi
