Foundations of Mathematics

Transcription

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2 Erwin Engeler Foundations of Mathematics Questions of Analysis, Geometry & Algorithmics Translated by Charles B. Thomas With 29 Figures Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest

3 Author: Erwin Engeler Mathematikdepartment ETH-Zentrum CH-8092 Zurich, Switzerland Translator: Charles B. Thomas DPMMS, 16 Mill Lane Cambridge CB2 ISB, Great Britain Title of the original German edition: Metamathematik der Elementarmathematik in the Series Hochschultext 1983 by Springer-Verlag Mathematics Subject Classification (1991): 03-XX ISBN-13: e-isbn-13: DOl: / Library of Congress Cataloging-in-Publication Data Engeler, Erwin. [Metamathematik der E1ementarmathematik. English] Foundations of mathematics: questions of analysis, geometry & algorithmics / Erwin Engeler ; translated by Charles B. Thomas. p. cm. Includes bibliographical references. I. Metamathematics. I. Title. QA9.8.E O'.I-<lc This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg 1993 Sof'tcover reprint of the hardcover I st edition 1993 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready by author/translator using Springer TEX macropackage 41/

4 Preface This book appeared about ten years ago in Gennan. It started as notes for a course which I gave intermittently at the ETH over a number of years. Following repeated suggestions, this English translation was commissioned by Springer; they were most fortunate in finding translators whose mathematical stature, grasp of the language and unselfish dedication to the essentially thankless task of rendering the text comprehensible in a second language, both impresses and shames me. Therefore, my thanks go to Dr. Roberto Minio, now Darmstadt and Professor Charles Thomas, Cambridge. The task of preparing a La'JEX-version of the text was extremely daunting, owing to the complexity and diversity of the symbolisms inherent in the various parts of the book. Here, my warm thanks go to Barbara Aquilino of the Mathematics Department of the ETH, who spent tedious but exacting hours in front of her Olivetti. The present book is not primarily intended to teach logic and axiomatics as such, nor is it a complete survey of what was once called "elementary mathematics from a higher standpoint". Rather, its goal is to awaken a certain critical attitude in the student and to help give this attitude some solid foundation. Our mathematics students, having been drilled for years in high-school and college, and having studied the immense edifice of analysis, regrettably come away convinced that they understand the concepts of real numbers, Euclidean space, and algorithm. The advocacy of a critical attitude towards time-honoured teachings does not mean a denial of the usefulness, let alone the necessity, of a general consensus in mathematics. What I vehemently oppose is the lack of imagination that underlies the tendency to blind acceptance and thoughtless compliance with which many students approach the basic attitudes and concepts of mathematics. To combat this tendency, mathematical logic has created technical machinery, in terms of which such a critique can be cogently, in fact mathematically, expressed. The word "metamathematics" in the original Gennan title of this book thus indicates the method; "elementary mathematics" indicates the subject area of the critique, namely, analysis, geometry and algorithmics. In a nutshell: how does one arrive at the axioms of elementary mathematics and what do we gain by having them? Zurich, September 1992 E. Engeler

5 Contents Chapter I. The Continuum What Are the Real Numbers? Language as Part of Mathematics Elementary Theory of Real Numbers N on-standard Analysis Axiom of Choice and Continuum Hypothesis Chapter II. Geometry Space and Mathematics Axiomatization by Means of Qoordinates Metatheoretical Questions and Methods in Elementary Geometry Geometric Constructions Chapter III. Algorithmics What is an Algorithm? The Existence of Combinatory Algebras: Combinatory Logic Concrete Combinatory Algebras Lambda Calculus Computability and Combinators... 94