Real Numbers, Generalizations of the Reals, and Theories of Continua
Since their appearance in the late 19th century, the Cantor--Dedekind theory of real numbers and philosophy of the continuum have emerged as pillars of standard mathematical philosophy. On the other hand, this period also witnessed the emergence of a vari
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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 ofGroningen, The Netherlands PATRICK SUPPES, Stanford University, California JAN WOLENSKI, Jagiellonian University, Krak6w, Poland
VOLUME 242
REAL NUMBERS, GENERALIZATIONS OF THE REALS, AND THEORIES OF CONTINUA Edited by
PHILIP EHRLICH Ohio University
Springer-Science+Business Media, B. V.
Library of Congress Cataloging-in-Publication Data
Real numbers, general lzatlons of the reals, and theorles of contlnua I edited by Phl11p Ehrlich. p. c •. -- (Synthese l1brary ; v, 242) Includes index. 1. Numbers, Real. Ir. Serles.
2. Continuum hypothesis.
1. Ehrlich, Phillp.
OA241.R34 1994 512' .7--dc20
93-47519
ISBN 978-90-481-4362-7 ISBN 978-94-015-8248-3 (eBook) DOI 10.1 007/978-94-015-8248-3
All Rights Reserved © 1994 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1994. Softcover reprint of the hardcover I st edition 1994 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 permission from the copyright owner.
TABLE OF CONTENTS
PHILIP EHRLICH / PART I.
General Introduction
vii
THE CANTOR-DEDEKIND PHILOSOPHY AND ITS EARLY RECEPTION
On the Infinite and the Infinitesimal in Mathematical Analysis (Presidential Address to the London Mathematical Society, November 13, 1902)
E. W. HOBSON /
PART II.
3
ALTERNATIVE THEORIES OF REAL NUMBERS
DOUGLAS S. BRIDGES /
Number Line
J. H. CONWAY /
A Constructive Look at the Real
The Surreals and Reals
29 93
PART III. EXTENSIONS AND GENERALIZATIONS OF THE ORDERED FIELD OF REALS: THE LATE 19TH-CENTURY GEOMETRICAL MOTIVATION GORDON FISHER /
Continuum
Veronese's Non-Archimedean Linear
Review of Hilbert's Foundations of Geometry (1902): Translated for the American Mathematical Society by E. V. Huntington (1903) GIUSEPPE VERONESE / On Non-Archimedean Geometry. Invited Address to the 4th International Congress of Mathematicians, Rome, April 1908. Translated by Mathieu Marion (with editorial notes by Philip Ehrlich) HENRI POINCARE /
PART IV.
107 147
169
EXTENSIONS AND GENERALIZATIONS OF THE REALS: SOME 20TH-CENTURY DEVELOPMENTS
HOUR Y A SIN ACEUR /
Calculation, Order and Continuity v
191
vi
TABLE OF CONTENTS
H. JEROME KEISLER / The Hyperreal Line PHILIP EHRLICH / All Numbers Great and Small DIETER KLAUA / Rational and Real Ordinal Numbers
207 239 259
INDEX OF NAMES
277
PHILIP EHRLICH
GENERAL INTRODUCTION
The geometers of ancient Greece regarded number as a "multitude composed of units" (Euclid, p. 277) and, believing that one was not itself a number, but rather the unit or source of number, tended to identify the numbers with the posit
