Truth, Proof and Infinity A Theory of Constructions and Constructive
Constructive mathematics is based on the thesis that the meaning of a mathematical formula is given, not by its truth-conditions, but in terms of what constructions count as a proof of it. However, the meaning of the terms `construction' and `proof' has n
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SYNTHESE LIBRARY STUDIES IN EPISTEMOLOGY, LOGIC, METHODOLOGY, AND PHILOSOPHY OF SCIENCE
Managing Editor: JAAKKO HINTIKKA, Boston University
Editors: DIRK VAN DALEN, University of Utrecht, The Netherlands DONALD DAVIDSON, University of California, Berkeley THEO A.F. KUIPERS, University ofGraningen, The Netherlands PATRICK SUPPES, Stanford University, California JAN WOLEN-SKI, Jagielionian University, KrakOw, Poland
VOLUME 276
PETER FLETCHER Department of Mathematics, Keele University, United Kingdom
TRUTH, PROOF AND INFINITY A Theory of Constructions and Constructive Reasoning
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-5105-9 ISBN 978-94-017-3616-9 (eBook) DOI 10.1007/978-94-017-3616-9
Printed on acid-free paper
AII Rights Reserved © 1998 Springer Science+-Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1998 Softcover reprint of the hardcover 1st edition 1998 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner
TABLE OF CONTENTS
PREFACE .................................................... vii PART I: Philosophical Foundations ................................ 1 1. Introduction and Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . 1 2. What's Wrong with Set Theory? ............................... 13 3. What's Wrong with Infinite Quantifiers? ........................ 24 4. Abstraction and Idealisation ................................... 41 5. What are Constructions? ...................................... 51 6. Truth and Proof of Logical Formulae ........................... 70 7. The Need for a Theory of Constructions ......................... 77 8. Theories of Constructions ..................................... 82 9. Hilbert's Formalism .......................................... 98 10. Open-endedness ........................................... 117 11. Analysis ................................................. 128 PART II: The Theory of Constructions ........................... 140 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.
Introduction to Part II ...................................... 140 Design of the Term Language ............................... 142 The Term Language ........................................ 150 From the Term Language to the Expanded Term Language ..... -~162 The Expanded Term Language .............................. 199 The Protological Sequent Calculus ........................... 209 Commentary on the Protological Axioms and Rules ............. 212 From Protologic to Expanded Protologic ...................... 216 Expanded Protologic ....................................... 231 From Expanded Protologic to the Coding of Trees .............. 234 The Coding of Trees ....................................... 253 The
