Principles of Intuitionism Lectures presented at the summer conferen
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95
A. S. Troelstra University of Amsterdam, Amsterdam
Principles of Intuitionism Lectures presented at the summer conference on Intuitionism and Proof theory (1968) at SUNY at Buffalo, N.Y.
1969
Springer-Verlag Berlin · Heidelberg· New York
Lecture Notes in Mathematics A collection of informal reports andseminars Edited by A. Dold, Heidelberg and B. Eckmann, ZUrich
95
A. S. Troelstra University of Amsterdam, Amsterdam
Principles of Intuitionism Lectures presented at the summer conference on Intuitionism and Proof theory (1968) at SUNY at Buffalo, N.Y.
1969
Springer-Verlag Berlin · Heidelberg· New York
All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer Verlag. © by Springer-Verlag Berlin· Heidelberg 1969 Library of Congress Catalog Card Number 74 -88182 Printed in Germany. Title No. 3701
Contents 1. Int roduct ion ..•..................••......•........... ,. . . . • . • .
2
2. Logic ••....•...•.......••••.•..••••••••.••••••••.•••••••••••
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3. 4. 5. 6.
Elementary arithmetic •••••••••••••••••••••••••••••••••••••••• 12 Species ••.•••.•.....••.•....•••••.•••.•.•.•..••••...••••.•••• 14
Sequences and lawlike •••••••••••••••••••••••••••••••• Elementary theory of real numbers •••••••••••••••••••••••••••• 7. Ordering relations and order on the real line ••••••••••••••••• 8. Constructive or lawlike analysis ••••••••••••••••••••••••••••• 9. Lawless sequences of natural numbers •••••••••••••••••••••••••
16 22 26
29 34
10. Choice sequences •••••••••••••••••...••••••••••••••••••••••••• 44
11. Spreads and the theory of real numbers •••••••••••••••••••••••• 57 12. Topology; separable metric spaces •••••••••••••••••••••••••••• 64
13. Applications of the continuity principles and the fan theoremy 71 14. Well-orderings and ordinals •••••••••••••••••••••••••••••••••• 76 15. Species revisited; the r;le of the comprehension principle 91 16. Brouwer's theory of the creative subject ••••••••••••••••••••• 95 17. Bibliography •••••••••••••••••••••••••••••••••••••••••••••••••108
- 2 § 1. Introduction
1.1 This paper is intended as an introduction to the principles of intuitionism. Basic notions, not formal systems are emphasized; proof theoretic results are mentioned to illustrate relative power and formal consequences of various principles. In other words, this paper presents the material needed to recognize certain formal systems as representing fragments of intuitionistic mathematics. The application of various principles in mathematics is illustrated by suitable examples. Although the paper has more or less a survey character, it is not exhaustive; various important subjects are summarily treated or mentioned only (e.g. G5del's Dialectica interpretation and functionals of higher type, completeness problems of intuitionistic logic). The selection of the examples from mathematics was determined by their usefulness as illustrations and by personal preference; thus many well-developed subjects of intuitionistic mathematics are not touched u
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