Real Numbers, Generalizations of the Reals, and Theories of Continua

REAL NUMBERS, GENERALIZATIONS OF THE REALS, AND THEORIES OF CONTINUA

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 of Groningen, The Netherlands PATRICK SUPPES, Stanford University, California JAN WOLETýSKI, JagiellonianUniversity, Krak6w, Poland

VOLUME 242

REAL NUMBERS, GENERALIZATIONS OF THE REALS, AND THEORIES OF CONTINUA Edited by

PHILIP EHRLICH Ohio University

KLUWER ACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON

Library of Congress Cataloging-in-Publication Data

Real numbers, generalizations of the reals, and theories of continue / edited by PhIlip Ehrlich. p. cm. -- (Synthese library ; v. 242) Includes index. ISBN 0-7923-2689-X (acid-free) 1. Numbers, Real. 2. Continuum hypothesis. I. Ehrlich, Philip. II. Series. CA241.R34 1994 512'.7--dc2O 93-47519

ISBN 0-7923-2689-X

Published by Kluwer Academic Publishers, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. Kluwer Academic Publishers incorporates the publishing programmes of D. Reidel, Martinus Nijhoff, Dr W. Junk and MTP Press. Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publishers, 101 Philip Drive, Norwell, MA 02061, U.S.A. In all other countries, sold and distributed by Kluwer Academic Publishers Group, P.O. Box 322, 3300 AH Dordrecht, The Netherlands.

All Rights Reserved © 1994 Kluwer Academic Publishers 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. Printed in the Netherlands

TABLE OF CONTENTS

PHILIP EHRLICH

/ General Introduction

vii

THE CANTOR-DEDEKIND PHILOSOPHY AND

PART I.

ITS EARLY RECEPTION HOBSON / On the Infinite and the Infinitesimal in Mathematical Analysis (Presidential Address to the London Mathematical Society, November 13, 1902)

E. W.

ALTERNATIVE THEORIES OF REAL NUMBERS

PART II. DOUGLAS S.

BRIDGES

/ A Constructive Look at the Real

Number Line J. H. CONWAY

PART III.

3

/ The Surreals and Reals

29 93

EXTENSIONS AND GENERALIZATIONS OF THE

ORDERED FIELD OF REALS: THE LATE 19TH-CENTURY GEOMETRICAL MOTIVATION GORDON

FISHER

/ Veronese's

Non-Archimedean Linear 107

Continuum / 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

POINCARt

PART IV.

147

169

EXTENSIONS AND GENERALIZATIONS OF THE REALS: SOME 20TH-CENTURY DEVELOPMENTS

HOURYA SINACEUR

/ Calculation, Order and Continuity v

191

vi

TABLE OF CONTENTS

/ The Hyperreal Line / All Numbers Great and Small

H. JEROME KEISLER

PHILIP EHRLICH DIETER KLAUA

/ Rational and Real Ordinal Numbers

INDEX OF NAMES

207 239 259 277

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 positive integers greater than one. The early modern theory of real numbers began to emerge during the latter part of the 16th century when mathematicians like Simon Stevin (1585) argued that not only is one also a number, but there is a complete correspondence between (positive) number and continuous magnitude, as well as a parallelism between certain geometrical constructions and the now familiar arithmetic operations on numbers. This point of view soon led to, and was implicit in, the analytic geometry of Descartes (1637), and was made explicit by John Wallis (1655) and Newton (1684) in their arithmetizations thereof. Following Wallis (1657), the (positive) numbers came to be associated with the ratios which were assumed to exist between the magnitudes of a given kind and a selected unit magnitude of the same kind, where in accordance with Euclid and Eudoxus, magnitudes were understood "to have a ratio to one another which are capable, when multiplied [by a positive integer] of exceeding one another" (Euclid, p. 114). Wallis further supposed that no two magnitudes of the same kind could differ by an infinitesimal amount, and the numbers were said to be either rational (whole or broken) or irrational depending upon whether or not the magnitudes in question are commensurable or incommensurable with the given unit. In his Arithmetica Universalis (1684), Newton extended the correspondence between numbers and ratios to include negative numbers and zero, but whereas Wallis identified the positive numbers with the symbolic representations of ratios, Newton identified numbers with the 'abstracted' ratios themselves. The fact that zero could not be a number in accordance with his definition did not preclude Newton from asserting it was (1684, p. 7); and the careful treatment required to handle the ratios of his directed magnitudes is nowhere to be found. Prior to the arithmetization of analysis, there was an implicit distinction drawn by many analysts between continuous Euclidean vii P. Ehrlich (ed.), Real Numbers, Generalizationsof the Reals, and Theories of Continua, vii-xxxii. © 1994 Kluwer Academic Publishers. Printed in the Netherlands.

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(geometrical) magnitude and the continuous magnitude of analysis, the latter of which was assumed to be richer than the former. Unlike continuous Euclidean magnitude, which was the basis of the real number concept, the continuous magnitude of analysis was thought to require an appeal to infinitesimal quantities of one sort or another for its understanding. However, by the second half of the 19th century a growing number of mathematician viewed both conceptions with mistrust. Developments in analysis persuaded them that the traditional notions were much two imprecise, unreliable and ineffective to provide a rigorous foundation for analysis, and the historical bifurcation of continua appeared to be more of a hinderance than an aid in achieving this end. In response to this state of affairs, the modem Arithmetico-set-theoretical conception of a real number emerged during the latter half of the 19th century when a number of mathematicians including Cantor (1872) and Dedekind (1872) introduced theories of real numbers that were designed to be independent of the former notion, and intended to make possible an analysis which banished the latter. The newly constructed ordered field of real numbers was dubbed the arithmetic continuum because it was held that this number system is completely adequate for the analytic representation of all types of continuous phenomena. In accordance with this view, the geometric linear continuum was assumed to be isomorphic with the arithmetic continuum, the axioms of geometry being so selected to insure this would be the case. In honor of Cantor and Dedekind, who first proposed the thesis, the presumed correspondence between the two structures has come to be called the Cantor-Dedekind axiom. Given the Archimedean nature of the real number system, once this axiom is adopted we have the classic result that infinitesimal line segments are superfluous to the analysis of the structure of a continuous straight line. Since their appearance, the late 19th-century constructions of real numbers have undergone substantial and much needed logical and settheoretical refinement, and the systems of rational and integer numbers upon which they are based have themselves been given a set-theoretic foundation. During this period the Cantor-Dedekind philosophy of the continuum has also emerged as a pillar of standard mathematical philosophy and it currently underlies the standard formulation of the calculus, the standard analytic and synthetic theories of geometrical linear continua, and the standard axiomatic theories of continuous magnitude more generally. On the other hand, this period has also witnessed the

GENERAL INTRODUCTION

ix

emergence of the theories of non-Archimedean ordered algebraic and geometric systems, Non-Standard Analysis (Robinson, 1961; 1966) and Smooth Infinitesimal Analysis (Moerdijk and Reyes, 1991), as well as

a variety of alternative theories of real numbers and/or alternative theories of continua. A number of important generalizations of the system of real numbers have also appeared, some of which have been described as arithmetic continua of one type or another. With the exception of the opening essay by Hobson, which is primarily concerned with the ideas

of Cantor and Dedekind and their reception at the turn of the century, the papers in the present collection are either concerned with, or are contributions to, the latter groups of studies. In what follows, we briefly motivate, describe and comment on the essays. Somewhat more attention will be paid to motivating the papers in Part III, not because we believe that these papers are more impor-

tant than the others, but rather because the subject of non-Archimedean geometry with which they are concerned is far less well-known. A

substantially expanded discussion of the material contained in that section will be found in a forthcoming paper by the editor. PART I.

THE CANTOR-DEDEKIND PHILOSOPHY AND ITS EARLY RECEPTION

Although the Cantor-Dedekind theory of real numbers and philosophy of the continuum occupy privileged positions in present day mathematical philosophy, in the decades preceding the turn of the century they had not yet become so well entrenched. Many of the works of applied mathematicians and number of textbooks in common use (particularly in England and America) were written as if the arithmetizing school did not exist; and the ferment and clash of opposing views which is characteristic of paradigm shifts was in the air. Hankel, himself a creator of an arithmetical theory of rational numbers (1867), expressed strong opposition to Weierstrass' early call for the complete separation of number and magnitude when he wrote: Every attempt to treat the irrational numbers formally and without the concept of magnitude must lead to the most abstruse and troublesome artificialities, which, even if they can be carried through with complete rigor, as we have every right to doubt, do not have a higher scientific value. (1867, p. 46)

And some mathematicians, like Paul du Bois-Reymond, while embracing

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PHILIP EHRLICH

many of the methods of the arithmetical school, also continued to regard the idea of magnitude as central to the foundation of the number concept. Indeed, warning against the new approach he wrote: No doubt, with help from so-called axioms, from conventions, from philosophemes contrived ad hoc, from unintelligible extensions of originally clear concepts, a system of arithmetic can be constructed which resembles in every way the one obtained from the

concept of magnitude...

But: A purely formalistic-literal framework of analysis which is what the separation of number from magnitude amounts to, would degrade this science to a mere game of symbols... (1882, p. 53)

Du Bois-Reymond's critique, however, was not restricted to the separation of number and magnitude. While not sharing the qualms about actual infinities or non-denumerable infinities which would be characteristic of many of the early 20th-century intuitionists and semiintuitionists, respectively, he anticipated one of their central concerns when he attacked the Cantor-Dedekind philosophy of the continuum on the ground that it was committed to the reduction of the continuous to the discrete, a program whose philosophical cogency, and even logical consistency, had been challenged many times over the centuries. Thus he wrote: The conception of space as static and unchanging can never generate the notion of a sharply defined, uniform line from a series of points however dense, for after all, points are devoid

of size, and hence no matter how dense a series of points may be, it can never become an interval, which always must be regarded as the sum of intervals between points. (1882, Vol. 1, p. 66)

Du Bois-Reymond was not alone among late 19th-century thinkers in believing that, if a continuous line is to be regarded as composed of elements, these elements must themselves be extended; and this view gave rise to a number of alternative theories of continua based on one or another conception of infinitesimal. The most highly developed theory of this kind is the one worked out by G. Veronese in his pioneering work on non-Archimedean geometry (1891), and though not very well developed, the ideas of C. S. Peirce also deserve mention in this regard (cf. 1898; 1900; and Eisele, 1976). However, while the ideas of Cantor and Dedekind were still the source of considerable controversy and had not been universally embraced, by 1902 they had made substantial inroads among pure mathematicians

GENERAL INTRODUCTION

xi

and were beginning to assume a position of dominance. With the hope of further securing, not only their position but the position of the arithmeticizing school more generally, E. W. Hobson devoted his Presidential Address to the London Mathematical Society that year to a discussion of these matters. This now largely forgotten address, entitled The Infinite and Infinitesimal in Mathematical Analysis serves as the opening contribution to the present collection. In it the reader will find not only an elegant statement of the Cantor-Dedekind philosophy in the context of a historical discussion of the broader topic expressed in the paper's title, but a sensitive analysis which beautifully conveys the revolutionary and unsettled character of the time. PART II.

ALTERNATIVE THEORIES OF REAL NUMBERS

In the ensuing years, the position of the Cantor-Dedekind philosophy became increasingly more solidified. On the other hand, paradigms rarely, if ever, enjoy the complete allegiance of their respective communities, and the Cantor-Dedekind philosophy is no exception in this regard. Indeed, up to the present day there has never been a time at which it has met with the universal acceptance of either philosophers or mathematicians, and the challenges that have been directed towards it have been varied and represent a wide variety of mathematico-philosophical perspectives. However, of all its critics perhaps the most persistent have been the constructivists. 'Constructivism' is a rubric that has come to designate a heterogeneous family of schools that are loosely united by their opposition to certain forms of mathematical reasoning employed by the mainstream mathematical community. Included among these schools are those of Finitism, Predicativism,Intuitionism, Constructive Recursive Mathematics, Bishop's ConstructiveMathematics and Semi-intuitionism. As the term 'loosely' suggests, however, there are important differences between the various schools; and, in fact, substantial differences in attitude can be found, even among the representatives of a given school or a single representative over time. However, whether the result of the rejection of actual infinities or nondenumerable infinities, or the insistence upon the use of predicative definitions or algorithmic constructions of one form or another, constructivists have always found themselves at odds with the standard analysis of continua and the corresponding theories of real numbers. In

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fact, to great extent, one may classify the various constructivist approaches to the foundations of mathematics according to the kind of non-standard theories of real numbers and continua they accept (cf. Fraenkel and Bar-Hillel, 1973, Ch. IV; Troelstra and van Dalen, 1988). Until the late 1960s the constructivist theory of continua which received the most attention from constructive mathematicians is 'the one' based on the idea of freely proceeding infinite sequences due to L. E. J. Brouwer (cf. van Dalen, 1981). However, despite the elegance and subtlety of the theory, it did not attract much attention from standard mathematicians. Whether this is because of the philosophical precepts underlying it, the highly non-classical nature of the mathematical arguments it employs, or the belief that the resulting mathematics is too impoverished, is difficult to say. In 1967, however, Brouwer's theory was given a particularly stinging critique, not by standard mathematicians, but by Errett Bishop, who is widely credited as being the author who breathed new and vibrant life into constructive mathematics. In his polemical manifesto on constructive mathematics Bishop characterized the construction and motivation underlying Brouwer's theory of continua in the following terms: Brouwer became involved in metaphysical speculation by his desire to improve the theory of the continuum. A bugaboo of both Brouwer and the logicians has been compulsive speculation about the nature of the continuum. In the case of the logicians this leads to contortions in which various formal systems, all detached from reality, are interpreted within one another in the hope that the nature of the continuum will somehow emerge. In Brouwer's case there seems to have been a nagging suspicion that unless he personally intervened to prevent it, the continuum would turn out to be discrete. He therefore introduced the method of free-choice sequences for constructing the continuum, as a consequence of which the continuum cannot be discrete because it is not well enough defined. This makes mathematics so bizarre it becomes unpalatable to mathematicians, and foredooms the whole of Brouwer's program. This is a pity, because Brouwer had a remarkable insight into the defects of classical mathematics, and he made a heroic attempt to set things right. (1967, p. 9)

In his treatise Foundations of Constructive Analysis (1967) Bishop attempted to place analysis on a constructive foundation that was free of the perceived difficulties referred to above. Unlike Brouwer's intuitionistic mathematics, which is incompatible with classical mathematics, Bishop's approach provides a generalization of the classical theory in much the same sense that the theory of groups is a generalization of the theory of ordered groups; thus, every proof in Bishop's theory is a classical proof and every theorem of Bishop's theory has an immediate

GENERAL INTRODUCTION

xiii

interpretation in classical mathematics. The converse, however, is not the case and this results in a markedly different theory of the real number system. During the past two decades Bishop has attracted a number of mathematicians who are constructivizing mathematics along the lines he set forth. Prominent among them is Douglas Bridges who co-authored Constructive Analysis (1985), which is the '2nd edition' of Bishop's classic work. In his contribution to the present collection, Bridges provides a detailed analysis of the real number line within the constructive framework erected by Bishop. The first three sections cover basic constructive material on sets, functions, logic, and the axioms of choice. Sections 4-10 describe the construction, and the fundamental algebraic, topological, and order properties, of the set of constructive real numbers. Section 11 deals with a famous theorem of Specker, giving a strong recursive counterexample to the classical least-upper-bound principle. The remaining three sections turn back to Bishop's real number line, and culminate in a detailed examination of various classically equivalent, but constructively distinct, notions of connectedness. Unlike the constructivists, there have been and continue to be critics of the Cantor-Dedekind theory of real numbers who are nevertheless sympathetic to both the Cantor-Dedekind philosophy of the continuum and the tenets of classical mathematics more generally. One of the well-known representatives of this group is Bertrand Russell, whose searching critique (1903, Ch. XXXIV) helped place the late 19th-century constructions of real numbers due to Cantor, Dedekind and Weierstrass on logically sound set-theoretic foundations. Recently, however, the distinguished mathematician J. H. Conway, who may also be regarded as a member of this group, has raised serious questions about the virtues of even the sanitized constructions. The thrust of his doubts are contained in the following lengthy passage from his monograph On Numbers and Games (1976, pp. 25-27). Figure 1 shows the lattice of inclusions between the sets Z, Q, R of integers, rationals and reals, and the corresponding sets Z', Q` R' of positive R

IN I- Q \ Q+ , z

R+

Fig. 1.

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PHILIP EHRLICH

elements. [It does not matter very much whether we add here the number 0 or not.] We shall suppose Ze and its properties already known. Then one sees at once that there are several possible paths through the lattice from Z' to R. Some experience in teaching convinces one that there is a unique best possible path, which is not one that seems natural at first sight! For X = Z or Q or R we can proceed from X' by introducing ordered pairs (a, b) meaning a - b, and the equivalence relation (a, b) - (c, d) iff a + d = b + c. [The alternative of adding new elements 0 and -x(x E X+) leads to too much case-splitting.]

Similarly we can proceed from Z to Q or /Z to Q÷ by introducing ordered pairs (a, b) meaning a/b and the equivalence relation (a, b)

-

(c, d) iff ad = bc.

We proceed from Q to R or Q' to R÷ by the method of Dedekind sections, or that of Cauchy sequences. In practice the main problem is to avoid tedious case discussions. [Nobody can seriously pretend that he has ever discussed even eight cases in such a theorem - yet I have seen a presentation in which one theorem actually had 64 cases!] Now if we define R in terms of Dedekind sections in Q, then there are at least four cases in the definition of the product xy according to the signs of x and y. [And zero often requires special treatment!] This entails eight cases in the associative law (xy)z = x(yz) and strictly more in the distributive law (x + y)z = xz + yz (since we must consider the sign of x + y). Of course an elegant treatment will manage to discuss several cases at once, but one has to work very hard to find such a treatment. This discussion convinces me that if one is to use Dedekind sections then the best treatment does not use the branch of our lattice from Q to R, and so must be the unique shortest path passing through R+. This seems surprising, since the algebraic theory (introduction of negatives and inverses) should naturally be logically prior to the analytic (limits, etc.). [The reader should be cautioned about difficulties in regard to the construction of the reals as a particular case of the completion of a metric space. If we take this line, we plainly must not start by defining a metric space as one with a real-valued metric! So initially we must allow only rational values for the metric. But then we are faced with the problem that the metric on the completion must be allowed to have arbitrary real values! Of course, the problem here is not actually insoluble, the answer being that the completion of a space whose metric takes values in a field F is one whose metric takes values in the completion of F. But there are still sufficient problems in making this approach coherent to make one feel that it is simpler to first produce R from Q, and later repeat the argument when one comes to complete an arbitrary metric space, and of course this destroys the economy of the approach. My own feeling is that in any case the apparatus of Cauchy sequences is logically too complicated for the simple passage from Q to R - one should surely wait until one has the real numbers before doing a piece of analysis!] This discussion should convince the reader that the construction of the real numbers by any of the standard methods is really quite complicated.

In his contribution to the present collection, Conway further expounds on these difficulties and he also provides an overview of his novel

theory of real numbers which overcomes them, the details of which are

GENERAL INTRODUCTION

XV

contained in his monograph, mentioned above. Central to his approach is the fact that all of the real numbers emerge by means of a simple recursive procedure that is independent of arithmetic considerations. This permits him to handle the arithmetic all at once following their introduction. In a review of Conway's monograph for Mathematical Reviews, John Dawson commented on Conway's new approach to real numbers in the following terms. The new theory of real numbers ... is a profound and revolutionary contribution to the foundations of analysis. Indeed the reviewer believes that Conway's approach will eventually replace the traditional construction for the reals. Initially, however, the theory is likely to encounter resistance in many quarters, especially among traditional analysts less accustomed than logicians to the routine handling of inductive definitions and proofs. (Dawson, 1978, p. 1130)

Although only time can tell whether or not Dawson's prediction will be born out, I am confident that readers of Conway's essay and monograph will appreciate the basis of Dawson's enthusiasm. The Emergence of Non-Archimedean Mathematics Even before Cantor and Dedekind had published the modern theories of real numbers which would be employed to 'banish' infinitesimals from late 19th- and pre-Robinsonian, 20th-century analysis, Johannes Thomae (1870) and, particularly, Paul du Bois-Reymond (1870-1871) were beginning the process which would, in the years bracketing the turn of the century, not only establish a consistent and relatively sophisticated algebraic theory of infinitesimals in mainstream mathematics, but make it, and especially the closely related subject of non-Archimedean Geometry, the focal point of great interest and an intensive research program. Out of the same body of work of the early 1870s there also emerged a largely parallel development of du Bois-Reymond's Infinitdrcalcfil which led, in the same period, to the famous work of Hardy (1910); but this rather different approach to infinitesimals will only be of momentary interest to us here. By the early 1880s, du Bois-Reymond and Otto Stolz (1883, 1884)

had already introduced number systems and a fledgling theory thereof that Abraham Robinson aptly described as a modest but rigorous theory of non-Archimedean systems (1967, p.39).

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Du Bois-Reymond's system (of Orders of Infinity) emerged in connection with jiis work on the rate of growth offunctions, and Stolz's closely related system (of Moments) was introduced in a now little known paper which is of some significance for the history of non-standard analysis. Here we find an especially early example of a staunch champion of the modern limit approach to analysis who, while seeing no need for it, is willing to leave open the possibility that, somewhere down the road, there might be an equally satisfactory foundation based on infinitesimals. The number systems of du Bois-Reymond and Stolz acquired large audiences through their incorporation in Stolz's highly respected textbook Vorlesungen uber allgemeine Arithmetik (Lectures on General Arithmetic 1885). Their role in the textbook, however, has little to do with the rate of growth of functions or the calculus; rather, they are offered as examples of systems which, unlike the system of real numbers, fail to satisfy the Archimedean axiom. With his Lectures Stolz was thereby able to rapidly spread the word of his two important discoveries of 1881-1883: namely, (i) There are systems of magnitudes (i.e., ignoring minor subtleties, what we today call ordered Abelian groups and their nonnegative and positive cones) which are non-Archimedean; and (ii) Systems of magnitudes that are continuous in the sense of Dedekind are Archimedean. With these discoveries, Stolz laid the groundwork for the modern theory of magnitudes - the branch of late 19th- and early 20th-century mathematical philosophy which would, in the decades that followed, evolve into the more general theory of ordered algebraic systems; and his work was soon followed by contributions of Veronese (1889) and Bettazzi (1890), the latter of which won the then prestigious prize of the Accademia Dei Lincei. In addition to early contributions to the theory of non-Archimedean ordered Abelian groups and semigroups, Bettazzi's work also contains the first satisfactory proof of (ii) as well as the first proof that each Archimedean ordered Abelian group is isomorphic to an ordered group of real numbers. Sadly, however, like the works of Stolz, these historically important works are now all but forgotten, having been completely overshadowed by the important work of H6lder (1901) and the great and vastly expansive work of Hahn (1907). The non-Archimedean systems of magnitudes studies by Stolz, Veronese, and Bettazzi in the works just mentioned, are additive struc-

GENERAL INTRODUCTION

xvii

tures which sometimes have modest multiplicative structures as well. However, unlike the real number system, none of them is an ordered field, or anything close to it. Just as ordered fields of real numbers arose in conjunction with the study of Euclidean geometry, it was from the study of non-Archimedean geometry that non-Archimedean ordered fields emerged. It is with the origins and early development of this investigation that the papers in the third part of the collection are concerned. PART III.

EXTENSIONS AND GENERALIZATIONS OF THE ORDERED FIELD OF REALS: THE LATE 19TH-CENTURY GEOMETRICAL MOTIVATION

Since the time that Wallis and Newton incorporated directed segments into Cartesian geometry, it has been well-known (albeit not always with complete precision) that given a unit segment AB of a line L of a classical Euclidean space, the collection of directed segments of L emanating from A including the degenerate segment AA itself constitutes an Archimedean ordered field with AA and AB the additive and multiplicative identities of the field and addition and multiplication of segments suitably defined. However, it was not until after Pasch (1882) included the projective formulation of the Archimedean condition in his axiomatization of projective geometry and Stolz (1882, 1883) clarified the watershed nature of the Archimedean condition in the Eudoxean Theory of Proportionsthat the following ideas began to emerge: it is possible to construct an axiomatization for the central theorems of Euclidean geometry that is independent of the Archimedean axiom and for which the aforementioned system of line segments in a model of the geometry continues to be an ordered field; however, in those models of the geometry in which the Archimedean axiom fails, the ordered fields in question are non-Archimedean ordered fields. These important insights first occurred to Giuseppe Veronese and they are developed in his influential pioneering work Fondamenti di Geometria (1891; German translation 1894), where there is also a detailed parallel development of Riemannian geometry independent of the Archimedean condition. Veronese's construction of non-Archimedean ordered fields of line segments is clumsy and quite complicated, though to some extent the complexity is by-product of what he is attempting to achieve. Indeed, Veronese is not merely attempting to construct a non-Archimedean orderedfield of segments which is appropriate to the

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geometry in question but, moreover, an orderedfield of 'power series' of such segments which models his novel theory of non-Archimedean continua, and do all this (to the extent that it is possible) by quasi-constructivist synthetic geometrical means! There is also a geometrically motivated system of symbols (Veronese's Numbers) which is introduced for the purpose of arithmetic representation, but for the sake of brevity we will not discuss it here. The motivation underlying Veronese's gradually emerging synthetic 'power series' construction is to provide detailed insight into how a non-Archimedean generalization of the classical method of scale construction could proceed in principle. For this purpose one requires a relatively sophisticated theory of the infinitely large and the infinitely small - a theory which makes use (at least implicitly) of a whole array of group-theoretic and field-theoretic concepts such as Archimedean class (i.e., what Robinson later called a galaxy), factor group, residue class field, value group, pseudo-convergent sequence, pseudo-limit, and the like. Unlike Hilbert and most other geometers, Veronese was not willing to leave such matters to set-theoretically minded order-algebrists like Levi-Civita, Holder, Schoenflies, Hahn, Hausdorff, Baer, Artin, Schreier, Ostrowski and Krull. Indeed, it was precisely the desire to initiate and contribute to the development of such a theory which motivated much of his work. After all, argued Veronese, the question of the nature of the rectilinear continuum emerges from synthetic geometry, and it is therefore incumbent upon synthetic geometry to lead the way in revealing its structure; but since, contrary to the theories of Cantor and Dedekind, there is nothing in the concept of a rectilinear continuum which necessitates the satisfaction of the Archimedean condition, it follows that a synthetic development of such a theory is not only appropriate, but required, for an adequate solution to the great ancient problem. Despite the lack of elegance in its presentation and elements of obscurity in its formulation, the theory of rectilinear continua developed in the Fondamenti is a profound and relatively sophisticated scheme, several of whose central concepts and ideas permeate the 20th-century theory of ordered algebraic systems and through it non-standard analysis. It is formulated in terms of a wide array of definitions and hypotheses, including two purported continuity hypotheses, each of which, unlike the Dedekind continuity condition, is satisfiable by both Archimedean and non-Archimedean ordered groups and fields. In fact, as Veronese himself demonstrated, each of the conditions is equivalent to the Dedekind

GENERAL INTRODUCTION

xix

continuity condition if the Archimedean axiom is assumed, though as

Levi-Civita later showed (1898), they are independent otherwise. One of the two continuity conditions first appeared (in essentially the same form) in Veronese's aforementioned paper of 1889; this is Veronese's principle of absolute continuity. Veronese formulates the principle for an system of magnitudes I having certain properties exemplified by the system of non-directed segments of a Euclidean line

including strict positivity (x + y > x, y) for the members of I - {01

and the absence of a smallest positive element. In the case of structures like 1, assuming the absence of a smallest positive element is equivalent to assuming that whenever x < y there is a z such that x < z < y. Following a long string of definitions and subsequent elaborations (pp. 608-610), Veronese states the principle in the following manner which, though suggestive, requires some unpacking: If an interval (XX') whose extremities always vary in opposite directions becomes indefinitely small, it always contains an element Y of I distinct from X and X'. (1889, p. 612, Princ. IV)

If one simply replaces the references to the variables X and X'with the collections A and B of values the variables assume, then on the basis of Veronese's definitions we arrive at the following crisp formulation of the condition which was made popular by Hblder (1901, pp. 10- 11): If A and B are nonempty subsets of Y where A has no greatest member, B has no smallest member, and every member of A precedes every member of B, and if for each positive member c of Y there are elements a of A and b of B for which b - a < c, then there is a z in I lying

strictly between the members of A and those of B. Moreover, since the element z is unique (1889, p. 612), the condition can also be stated in the following form, which was made popular by Schoenflies (1906, p. 26) and which more clearly highlights its relation to the Dedekind continuity condition: If (A, B) is a Dedekind cut of Y such that for each positive member c of I there are elements a of A and b of B for which b - a < c, then

either A has a greatest member or B has a least member, but not both. It is a simple matter to show that in the Archimedean case, and only in the Archimedean case, Veronese's metrical condition on cuts is invari-

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ably satisfied and therefore superfluous. Thus, for Veronese, unlike Dedekind, continuous systems of magnitudes need not be completely devoid of gaps, although they must be devoid of those gaps which satisfy the metrical condition satisfied in the classical case. Moreover, as Veronese showed, if (A, B) is a cut in a non-Archimedean ordered field which satisfies this condition, then the differences between the members of B and those of A must become infinitesimally small relative to any given positive element of the field, where, following Veronese (and virtually all other mathematicians since), a positive element v is said to be infinitesimal relative to a positive element u if and only if nv < u for all positive integers n. Veronese's second continuity condition, which he calls the hypothesis of relative continuity, first appeared in the complicated framework of the Fondamenti (p. 128). However, since it can be applied to F with no serious loss of content, for the sake of brevity we will follow this course. In order to formulate the condition we require the concept of an Archimedean class of 1, that is, a subclass of 2 consisting of all elements x and y of Y_which are finite relative to one another, where, following Veronese, positive elements x and y are said to befinite relative to one another if and only if there are positive integers m and n such that mx > y and ny > x. The idea of an Archimedean class was introduced independently by Veronese and Bettazzi and the appellation 'Archimedean class', which is now standard in the theory of ordered algebraic systems, is due to Neumann (1949); making use of this terminology, Veronese's relative continuity condition can be stated as follows: If A and B are non-empty subsets of an Archimedean class F of Y where A has no greatest member, B has no smallest member, and every member of A precedes every member of B, and if for each member c of F there are elements a of A and b of B for which b - a < c, then there is a z in F lying strictly between the members of A and those of B. It is a simple matter to show that if Y consists of all the positive members of an ordered field, then assuming the satisfaction of the relative continuity condition is equivalent to assuming that the ordered field contains an isomorphic copy of the ordered field of real numbers. On the basis of this, one may prove a result which asserts, in effect, that if

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one limits one's attention to the members of r and ignores all infinitesimal differences which may be present, then the resulting system has the full structure of the positive cone of the ordered field of real numbers when +, ., and < are suitably defined. In essence, it was this result which Veronese took to be the ultimate import of the relative continuity condition. Before concluding our brief discussion of Veronese's continuity conditions, it should be noted that since the late 1940s these conditions, or equivalent variations thereof, have received a good deal of attention from mathematicians. This is particularly striking in the case of the absolute continuity condition which has proven to be of considerable interest to order-algebrists and some non-standard analysts. The interest of the algebrists derives, in large part, from the fact that every ordered group (resp. ordered field) has up to isomorphism a unique absolutely continuous completion which is itself an ordered group (resp. an ordered field). The completion arises in much the same manner as in the classical case. However, despite the fact that the condition was wellknown and widely discussed during the first decade of the 20th century under the rubric 'Veronese's continuity condition', with the exception of authors like Neumann (1949, p. 215) and Laugwitz (1975, p. 308), contemporary mathematicians tend to be unaware of its origin and the origin of the important kind of cuts it employs. In the recent literature these cuts are frequently called Dedekindean cuts, proper Dedekind cuts, Holder cuts or regularDedekind cuts, and Veronese's completeness condition is sometimes called Dedekindean completeness by algebrists and Scott completeness by logicians. The most thorough survey of the relevant ordered algebraic literature is by Priess-Crampe (1983, pp. 65-85, 129-149) and references to the important relevant works of Cohen and Goffman, Banazchewski, and Scott may be found in the editor's footnote [18] in Marion's translation of Veronese's paper contained in the present collection. Moreover, for an up-to-date exploration of some of the non-standard models of analysis that are absolutely continuous the reader should consult Keisler and Schmerl (1991). Outside of the mathematics community there is a widespread misconception which is typified by the following remarks: Using the tools of mathematical logic and model theory, Robinson succeeded in defining infinitesimals rigorously. (Dauben 1992, p. 76)

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. . . The German logician Abraham Robinson (1918-1974), who invented what is known as non-standardanalysis, thereby eventually conferred sense on the notion of an infinitesimal greater than 0 but less than any finite number. (Moore 1990, p. 69).

Robinson, however, was an authority on the theory of ordered algebraic systems before he became a non-standard analyst; and, the concept of an infinitesimal he employed in non-standard analysis is precisely the rigorous concept that has been employed by algebrists and geometers since the 1890s, i.e., the concept that goes back to Veronese (and others) and which entered the analytic theory of ordered fields when, at Veronese's request, Levi-Civita sought to provide Veronese's synthetic geometrical continuum with a logically sound analytic representation. Despite some historical omissions, this is all but acknowledged in Luxemburg's Introduction to the Selected Papers of Robinson on NonstandardAnalysis where he says: A very important aspect of nonstandard analysis hinted at earlier is that the new method does not consist merely in adding in a consistent way infinitesimals to the reals. This

had been done successfully. In the 1890s, Tullio Levi-Civita, responding to a question of Veronese concerning geometries, constructed a non-Archimedean totally ordered field whose elements are power series. (1979, p. xxxvi)

Levi-Civita's first construction of a non-Archimedean ordered field (1883) leads to a notational variant of the now familiar (lexicographically) ordered field of Laurent formal power series with coefficients and exponents in the reals. However, while this ordered field is continuous in the sense of Veronese and models Veronese's axioms of geometry, it only provides an analytic representation of a small subfield of Veronese's own synthetic continuum. Accordingly, Levi-Civita returned to the problem in his second paper on the subject (1898) where in addition to constructing an analytic model of Veronese's continuum, he introduced a more general construction which leads to a wide variety of absolutely continuous and Veronese continuous non-Archimedean ordered fields. Levi-Civita's latter construction of non-Archimedean ordered fields was soon followed by constructions by Hilbert (1899), Schoenflies (1906), and Hahn (1907). With some justification, mathematicians frequently identify Hahn's influential paper as the first truly major work in the theory of nonArchimedean ordered algebraic systems. In addition to Hahn's Completeness Theorem and Embedding Theorem, the latter of which has been described as "the deepest result in the theory of F. 0. [fully ordered]

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Abelian groups" (Fuchs, 1963, p. 56), it contains the just-mentioned construction of the celebrated Hahn Fields - every ordered field is isomorphic to a subfield of a Hahn field (cf. Conrad and Dauns, 1969). However, as Hahn was well aware, his own work was strongly influenced by Levi-Civita's, which in turn was strongly influenced by Veronese's synthetic geometrical ideas. However, despite the important influence Veronese's ideas have had on the development of non-Archimedean mathematics, his pioneering work on the subject is relatively unknown to philosophers, historians and mathematicians of our time. The aim of Fisher's and Marion's contributions to the present collection may be regarded as a first step towards remedying this state of affairs. Fisher's paper pays attention to some of the preliminary features of Veronese's system which lead up to his construction of the non-Archimedean linear continuum mentioned above. There is also a discussion of some of the philosophical underpinnings of Veronese's work which provides a useful supplement to Veronese's own discussion of these matters contained in his paper 'On Non-Archimedean Geometry' that has been translated for the present collection by Marion. In Appendices 2 and 3 of Fisher's work there are also translations of some remarks by Veronese on what he called the 'intuitive continuum', and of some of his opinions about the continuum of Cantor and Dedekind. Since Veronese's style is difficult (to say the least), to foster appreciation of his work, Fisher starts with a summary of the article by Hahn. Following the appearance of his Fondamenti di Geometria, a good portion of Veronese's continuing contribution to non-Archimedean Geometry consisted of defending and clarifying his revolutionary (and sometimes obscure and poorly stated) ideas against direct and indirect criticisms by authors such as Cantor, Peano, Vivanti, Killing, Klein and Schoenflies. Some of these critiques were constructive and ultimately supportive, and others, though negative were quite searching; still others were by and large reactionary, and at times even nasty. In addition to a variety of real or perceived shortcomings in the foundation and development of his theory, the critiques included challenges by Cantor, Peano, Vivanti, and Killing of the coherence of actual infinitesimal line segments; and denials by Cantor of, and related concerns of Killing about, the possibility of infinite numbers other than cardinals and ordinals. To some extent the aforementioned analytic work of Levi-Civita helped to silence some of his critics; but in certain quarters doubts about aspects of his work remained, and it was only with the publication of Hilbert's

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Grundlagen der Geometrie (1899) that the residual doubts about nonArchimedean geometry per se would fade. Within three years of the appearance of Hilbert's Grundlagen der Geometrie it had already appeared in both English and French; and in his review dated December 1, 1902, Veblen could already safely say: Since its appearance in 1899 Hilbert's work on The Foundations of Geometry has had a wider circulation than any other modern essay in the realms of pure mathematics (1903, p. 303).

Although it would be highly misleading to attribute the intense interest to a single aspect of the work, there is no question-judging from its reviews and the content of much of the literature it rapidly spawned that among the features that excited most, and most excited many, was not only Hilbert's relatively simple and elegant development of Euclidean geometry (including the theories of proportions and plane areas) independent of the Archimedean axiom, but his development and penetrating analysis of an entirely novel non-Archimedean geometry and a noncommutative non-Archimedean system of numbers which is required to model it. Indeed, whereas earlier generations had been captivated by the geometries of Lobachevsky and Riemann, "what seems to have struck Hilbert's contemporaries", as Nicholas Bourbaki puts it, "is 'nonArchimedean geometry"' (1968, p. 313). Among the reviewers of Hilbert's work who were clearly 'struck' by the importance of non-Archimedean geometry was Henri Poincar6, and outside of the early editions of the Grundlagen itself, perhaps no other work was more helpful in getting the message of its importance across than Poincer6's review (1902). Poincar6, after all, was widely regarded as the greatest mathematician of his day, and according to Poincar6, Hilbert's work on geometry, and in no small part non-Archimedean geometry, made the philosophy of mathematics take a long step in advance, comparable to those which were due to Lobachevsky, to Riemann, to Helmholtz, and to Lie. (1902, p. 272; 1903, p. 23)

The American Mathematical Society, in fact, placed such weight on Poincar6's review, that it decided to supplement the two detailed and very positive reviews of Hilbert's work it had already published (Sommer, 1900; Hedrick, 1902) with an English translation of Poincar6's (1903). And the review went on to draw additional attention to non-Archimedean

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geometry: first, when a longer version of it which took account of Hilbert's and Dehn's subsequent contributions to non-Archimedean geometry was published in 1905 in support of Hilbert's award of the prestigious Lobachevsky Prize and, again, when a large portion of it was incorporated in Poincard's widely reprinted report (1911; 1911a; 1912) supporting Hilbert's award of the even more prestigious Bolyai Prize.

Despite the substantial historical (and philosophical) import of Poincar6's review, it does not appear to be well known to contemporary readers. For this reason, and because in it readers will find both a good overview of Hilbert's earliest contribution to non-Archimedean geometry and a beautiful testament to the sense of excitement it caused, we feel certain that they will find its inclusion in the present collection particularly welcome. We do think it is important, however, to caution readers against taking Poincar6's review to be a work of historical scholarship. Unlike, say, Federigo Enriques' great Encyklopedia article Prinzipiender Geometrie

(1907; French translation 1911), it is not. Like the Grundlagen itself, Poincar6's review does not always give sufficient credit to, and in some cases does not even mention, the important contributions of others, a point which was made explicitly and implicitly by a number of writers of the time (cf. Veblen, 1903; Wilson, 1904, p. 77; and Enriques 1907). With respect to non-Archimedean geometry in particular Poincar6 makes no mention of the work of Veronese and Levi-Civita or, for example, Schur's (1898 [Preface]; 1899) own development of elementary geometry independent of the Archimedean axiom. We hasten to note, however, that at least in the case of Veronese, Poincar6 did attempt to make amends when he later added: In the article in the Bulletin that I devoted to the work of Mr. Hilbert on the foundations of Geometry, I particularly emphasized the importance of that new geometry which is named non-Archimedean. I should have added that the idea was first introduced by Mr. Veronese, and that Mr. Hilbert, while not making it, has given it a remarkably elegant form and better demonstrated its real range ... (1903a, p. 115; see, however, [Poincard 1905, p. 25] where Veronese is merely given the status of a 'precursor' and it is said that "Hilbert has really made this new geometry.")

xxvi PART IV.

PHILIP EHRLICH EXTENSIONS AND GENERALIZATION OF THE REALS: SOME 20TH-CENTURY DEVELOPMENTS

Following the works of Hilbert and Hahn, the next major development in the study of generalizations of the real number system was the creation of the modern theory of real-closed ordered fields, a theory which has fundamental significance for both Archimedean and non-Archimedean ordered fields as well as for arithmetic and geometric theories of continua. An ordered field may be said to be real-closed if it admits no algebraic extension to a more inclusive ordered field. Thus, real-closed ordered fields stand in the same relation to ordered fields that algebraicallyclosed fields bear to fields. Although the ordered field of real numbers is the most famous real-closed ordered field, it is by no means the only one. Indeed, for each ordered field there is up to isomorphism a smallest real-closed ordered field which contains it. The importance of real-closed ordered fields to the theory of elementary continua was greatly clarified when Tarski (1951) demonstrated that they are precisely the ordered fields which are first-order indistinguishable from the ordered field of reals, or, to put this another way, they are precisely the ordered fields which satisfy the first-order content of the Dedekind continuity axiom. Although ideas about real-closed ordered fields date back at least to the time of Lagrange, the modern theory was developed by Artin and Schreier (1926, 1927). Whereas Dedekind's theory of real numbers reduced continuity to order, Artin and Schreier's theory of real-closed ordered fields reduced orderability to calculation. Thus, when applied to the field of real numbers, Artin and Schreier's theory furnished the first completely algebraic theory of the linear continuum. In her contribution to the collection, Sinaceur recalls the historical background and the epistemological import of this theory; and she sketches in outline some of its contemporary applications. In the decades after the appearance of Artin and Schreier's work, the investigation of non-Archimedean number systems grew in a wide variety of directions including the further development of the theory of real-closed ordered fields. It was not until the early 1960s, however, that Abraham Robinson made the momentous discovery that among the real-closed extensions of the reals there are number systems which can provide the basis for a non-standard approach to analysis making use of infinitesimals (1961; 1966). These number systems which may be

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called Robinson extensions of the reals are indistinguishable from the reals in the following precise sense: let (R, S : S E F) be a relational structure where R is the set of real numbers and F is the set of all finitary relations defined on R. Also let *R be a proper extension of R and for each n-ary relation S E F let *S be an n-ary relation on *R that is an extension of S. The structure (*R, R, *S : S E F) may be said to be a Robinson extension of (R, S : S E F) if every n-tuple of real numbers satisfies the same first-order formulas in (R, S : S E F) as it satisfies in (*R, R, *S : S E F). The existence of Robinson extensions of the reals is a consequence of the compactness theorem of first-order logic and there are a number of algebraic techniques that can be employed to construct such extensions. One technique is the ultrapower construction, which is a special case of the ultraproduct construction. Ultraproducts of the reals were first studied (implicitly) by Hewitt (1948) and he called the real-closed extensions obtained this way hyperreal number systems. Log (1955) later isolated the concept of an ultrapower and showed (in a more general setting) that hyperreal number systems can be Robinson extensions of the reals. In his contribution to the present collection, Keisler explains what the hyperreal line is, what it looks like, and what it is good for. Near the beginning of the article he draws pictures of the hyperreal line and sketches its construction as an ultrapower of the real line. In the middle part, he surveys mathematical results about the structure of the hyperreal line, and near the end, he discusses philosophical issues concerning the nature and significance of the hyperreal line. I am confident that readers of Keisler's survey will find it to be among the most informed and up to date introductory discussions of this material in the literature today. Among the more important of the recent contributions to the theory of non-Archimedean ordered algebraic systems is the work of J. H. Conway on his number system No (1976; also see, Berlekamp et al., 1982; Gonshor, 1986; Ailing, 1987). No is up to isomorphism the unique real-closed ordered field such that for each pair X, Y of subsets of the field where every member of X precedes every member of Y, there is a member of the field lying strictly between those of X and those of Y (Ehrlich, 1988; 1989a). Since the latter property may be regarded as a condition of absolute density, No may be characterized up to isomorphism as the unique absolutely dense ordered field which admits no

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algebraic extension to a richer ordered field. In addition to the reals and the ordinals, No contains many less familiar numbers including (o/2, -ýo,1/0, and - (o to name only a few. In fact, subject to the proviso that numbers (construed here as members of ordered number fields) be individually definable in term of sets of standard set theory, it may be said to contain All Numbers Great and Small. Beside its sheer inclusiveness, which led Ehrlich (1986, 1989b, 1992) to argue that it could be regarded as a sort of absolute arithmetic continuum (modulo

standard set theory), two of the mathematically and philosophically most significant features of the structure are the existence of an algebraico-tree-theoreticsimplicity hierarchy among its numbers, and the

closely related fact that each number can be assigned its own 'proper name'. In his contribution to the present collection, Ehrlich provides an alternative construction of No based on a generalization of the von Neumann ordinal construction, and he uses it as a vehicle to bring attention to the simplicity hierarchy referred to above. Although Conway provided the first example of an ordered field that contains the entire class of ordinals, it was Sikorski (1948) who first made the important discovery that arbitrarily long initial sequences of ordinals can naturally arise in non-Archimedean ordered fields. This may come as a surprise to some readers, since ordered fields have a commutative structure, whereas (o + 1 # 1 + o) and (o-2 # 2.0o as we have all been taught. But the apparent inconsistency readily dissolves once it is realized that the sums and products of ordinals in ordered fields are not the familiar Cantorian sums and products, but rather the so-called natural

sums and products due to Hessenberg (1906). Sikorski showed that by starting with the set of all ordinals (written in Cantor Normal Form) less than a regular initial ordinal and employing the natural sums and products, one may obtain number systems that generalize the systems of integral and rational numbers by mimicking the familiar constructions of the integers from the natural numbers and the rationals from the integers. A little more than a decade later, Klaua (1956-1960, 1960) independently constructed these number systems which he called systems of integral and rational ordinal numbers and he extended the latter to

what he called systems of real ordinal numbers. To obtain the real ordinal numbers Klaua made use of a profound and original generalization of the Dedekind cut operation, and in the final paper of the collection he provides an overview of this intriging material and discusses some of

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the completeness properties of the number systems obtained in this fashion. Ohio University, Athens

Ohio, U.S.A. REFERENCES Ailing, N.: 1987, Foundations of Analysis Over Surreal Number Fields, North-Holland Publishing Co., Amsterdam. Artin, E. and Schreier, 0.: 1926, 'Algebraische Konstruktion Reeller Kdrper', Abhandlungen aus dem Mathematischen Seminar der Hamburgischen Universitlit, 5,

85-99. Artin, E. and Schreier, 0.: 1927, 'Eine Kennzeichnung der Reell Abgeschlossenen K6per', Abhandlungen aus dem Mathematischen Seminar der Hamburgishen Universitdt, 5,

225-231. Berlekamp, E. R. et al. 1982, Winning Ways for Your Mathematical Plays I, II, Academic

Press, New York. Bettazzi, R.: 1890, Teoria Delle Grandezze, Pisa. Bishop, E.: 1967, Foundations of Constructive Mathematics, McGraw-Hill, New York.

Bishop, E. and Bridges, D.: 1985, Constructive Analysis, Springer-Verlag, New York. Bourbaki, N.: 1968, Theory of Sets, Addison- Wesley Publishing Co., Reading, Massachusetts. Cantor, G.: 1872, 'Uber die Ausdehnung eines Satzes aus der Theories der Trigonometrischen Reihen', Mathematische Annalen, 5, 123-132.

Conrad, P. and Dauns, J.: 1969, 'An Embedding Theorem for Lattice-Ordered Fields', Pacific Journal of Mathematics, 30, 385-398.

Conway, J. H.: 1976, On Numbers and Games, Academic Press, New York. Dauben, J.: 1992, 'Appendix (1992): revolutions revisited' in D. Gillies (Ed.), Revolutions in Mathematics, Clarendon Press, Oxford, pp. 72-82. Dawson, J.: 1978, 'Review of J. H. Conway's On Numbers and Games', Mathematical Reviews, 56, 1129. Dedekind, R.: 1972, 1936, Essays on the Theory of Numbers, Dover, New York.

Du Bois-Reymond, P.: 1870-1871, 'Sur la grandeur relative des infinis des fonctions', Annali di Matematica pura de applicata (Series Ila), 4, 338-353. Du Bois-Reymond, P.: 1882, Allgemeine Functionentheorie,Tiobingen.

Ehrlich, P.: 1987, 'The Absolute Arithmetic and Geometric Continua', in A. Fine and P. Machamer (Eds.), PSA 1986, Vol. 2, Philosophy of Science Association, Lansing, MI, USA. Ehrlich, P.: 1988, 'An Alternative Construction of Conway's Ordered Field No', Algebra Universalis, 25, 7-16; Errata, 25, 233. Ehrlich, P.: 1989a, 'Absolutely Saturated Models', FundamentaMathematicae, 133, 39-46. Ehrlich, P.: 1989b, 'Universally Extending Continua', Abstracts of Papers Presented to the American Mathematical Society, 10 (January), 15.

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Ehrlich, P.: 1992, 'Universally Extending Arithmetic Continua', in H. Sinaceur and J. M. Salanskis (Eds.), Le Labyrinthe du Continu: Colloque du Cerisy, Springer-Verlag, France, Paris. Eisele, C. (Ed.): 1976, The New Elements of Mathematics by Charles S. Peirce, Vol. 3, Mouton, The Hague. Enriques, F.: 1907, 'Prinzipien der Geometrie' in Encykiopedia der Mathematischen Wissenschaften, Vol. III, pp. 1-129. Enriques, F.: 1911 a, 'Principes de la G6ometrie' in Encyclopidie des Sciences Mathdmatiques, Vol. III, pp. 1-147. Euclid: 1956, The Elements, Vol. II, T. Heath (Ed.), Dover, New York. Fraenkel, A. and Bar-Hillel, Y.: 1973, Foundations of Set Theory, North-Holland, New York. Fuchs, L.: 1963, Partially Ordered Algebraic Systems, Pergamon Press, New York. Gonshor, H.: 1986, An Introduction to the Theory of Surreal Numbers, Cambridge University Press, Cambridge. Hahn, H.: 1907, 'Uber die Nichtarchimedischen Grbssensysteme', Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften Wien: Mathematisch-Naturwissenschaftlichen Klasse Ha, 116, 601-655. Hankel, H.: 1867, Theorie der Komplexen Zahlsysteme, Leipzig. Hardy, G. H.: 1910, Orders of Infinity, Cambridge University Press, Cambridge; 2nd Ed., 1924. Hedrick, E.: 1902, 'The English and French Translations of Hilbert's Grundlagen der Geometrie', Bulletin of the American Mathematical Society, 9, 158-165. Hewitt, E.: 1948, 'Rings of Real-Valued Continuous Functions, I', Transactions of the American Mathematical Society, 64, 54-99. Hilbert, D.: 1899, Grundlagen der Geometrie, Leipzig. Hilbert, D.: 1900, 'Uber den Zahlbergriff', Jahresberichtder Deutschen MathematikerVereinigung, 8, 180-184. H61der, 0.: 1901, 'Der Quantitdt und die Lehre vom Mass', Berichte iiber die Verhandlungen der Ktiniglich Sdichsuschen GeseUschaft der Wissenschaften zu Leipzig, Matematisch-Physische Classe, 1-64. Keisler, H. J. and Schmerl, J.: 1991, 'Making the Hyperreal Line Both Saturated and Complete', Journal of Symbolic Logic, 56, 1016-1025. Klaua, D.: 1959-1960, 'Transfinite Reele Zahlenraume', Wissenschaftliche Zeitschrift der Humboldt- Universittit zu Berlin; Mathematische-NaturwissenshaftlicheReihe, 9, 169-172. Klaua, D.: 1960, 'Zur Structur der Reellen Ordinalzahlen', ZeitschriftfiirMathematische Logik und Grundlagen der Mathematik, 6, 279-302. Klein, J.: 1968, Greek Mathematical Thought and the Origins of Algebra, M.I.T. Press, Cambridge, MA. Laugwitz, D.: 1975, 'Tullio Levi-Civita's Work on Nonarchimedean Structures' in Tullio Levi-Civita Convegno Internazionale Celebrato Del Centenario Della Nascita, Accademia Nazionale dei Lincei, Atti dei Convegni Lincei, 8, 297-312. Levi-Civita, T.: 1893, 'Sugli Infiniti ed Infinitesmi Attuali Quali Elementi Analitici', in Opere Matematiche, Vol. I, Bolgna, 1954. Levi-Civita, T.: 1898, 'Sui Numeri Transfiniti', in Opere Mathematiche, Vol. 1, Bolgna 1954.

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Log, J.: 1955, 'Quelques Remargues, Th6ortmes, et Problkmes sur les Classes Ddfinissables d' Alg~bres', in T. Skokem et al. (Eds.), Mathematical Interpretation of Formal Systems, North-Holland, Amsterdam. Luxemburg, W. A. J.: 1979, 'Introduction to Papers on Nonstandard Analysis and Analysis', in H. J. Keisler et al. (Eds.), Selected Papers of Abraham Robinson, Vol. 2, Yale University Press, New Haven, 1979, pp. xxxi-xxxix. Moerdijk, I. and Reyes, G.: 1991, Models for Smooth Infinitesimal Analysis, SpringerVerlag, New York. Moore, A. W.: 1990, The Infinite, Routledge, London. Neumann, B. H.: 1949, 'On Ordered Division Rings', Transactions of the American Mathematical Society, 66, 202-252. Newton, I.: 1684, Arithmetica Universalis, in D. T. Whiteside (ed.), The Mathematical Works of Issac Newton, Vol. 2, 1967, Johnson Reprint Corporation, New York. Pasch, M.: 1882, Vorlesungen uber neuere Geometrie, Leipzig. Peirce, C. S.: 1898, 'The Logic of Continuity', in C. Hartshone and P. Weiss (Eds.), Collected Papers of Charles Sanders Peirce, Vol. VI, Harvard University Press, 1935. Peirce, C. S.: 1900, 'Infinitesimals', in C. Hartshone and P. Weiss (Eds.), Collected Papers of Charles Sanders Peirce, Vol. III, Harvard University Press, 1933. Poincard, H.: 1902, 'Hilbert's Grundlagen der Geometrie', Bulletin des sciences mathimatiques, 26me s~rie, 26, 249-272. Poincar6, H.: 1903 'Poincard's Review of Hilbert's Foundations of Geometry', Bulletin of the American Mathematical Society, Series 2, 10, 1-23. Poincar6, H.: 1905, 'Rapport sur les travaux de M. Hilbert', Bulletin De La Societd PhysicoMathimatique de Kasen, 2nd Series, 14, 10-48. Pioncar6, H.: 1911, Prix Bolyai 'Rapport de M. Henri Poincard', Bulletin des sciences mathdmatiques, 35, 67-100. Reprinted in Rendiconti Circolo Matematico Di Palermo, 31 (1911), 109-132; and in Acta Mathematica, 35 (1912), 1-28. Priess-Crampe, S.: 1983, Angeordnete Strukturen, Gruppen, Korper, Projektive Ebenen, Springer-Verlag, Berlin. Robinson, A.: 1961, 'Non-Standard Analysis', IndagationesMathematicae, 23, 432-440. Robinson, A.: 1966 (1974, 2nd ed.), Non-Standard Analysis, North-Holland, Amsterdam. Robinson, A.: 1967, 'The Metaphysics of the Calculus', in 1. Lakatos (Ed.), Problems in the Philosophy of Mathematics, North-Holland Publishing Co., Amsterdam. Russell, B.: 1903, Principles of Mathematics, Cambridge University Press (Norton), Cambridge. Schoenflies, A.: 1906, 'Uber die Mbglichkeit einer projektiven Geometrie bei transfiniter (Nicht archimedischer) Massbestimmung', Jahresbericht der Deutschen Mathematiker-Vereinigung, 15, 26-47. Schur, F.: 1898, Lehrbuch der analytischen Geometrie, Viet & Co., Leipzig. Schur, F.: 1899, 'Ueber den Fundamentalsatz der projectiven Geometrie, Mathematische Annalen, 51, 401-409. Sikorski, R.: 1948, 'On an Ordered Algebraic Field', La Sociiti des Sciences et des Lettres de Varsovie; Comptes Rendus des Siances de la Classe III, Sciences Mathimatiques et Physiques, 41, 69-96. Sinaceur, H.: 1991, Corps et Mod~les: Essai sur ' Historie de I' Algebra Rgele, Libraire Philosophique J. Vrin, Paris.

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Sommer, J.: 1900, 'Hilbert's Foundations of Geometry', Bulletin of the American Mathematical Society, Series 2, 6, 287-299. Stevin, S.: 1585, Arithmetique, in D. J. Struik (Ed.), The Principal Works of Simon Stevin, Vol. II B, 1958, C. V. Swets & Zeitlinger, Amsterdam. Stolz, 0.: 1881, 'B. Bolzano's Bedeutung in der Geschichte der Infinitesimalrechnung', Mathematische Annalen, 18, 255-279. Stolz, 0.: 1882, 'Zur Geometrie der Alten, insbesondere uber ein Axiom des Archimedes' Berichte des Naturwissenschaftlich-MedizinischenVereines in Innsbriick, 12, 74-89. Stolz, 0.: 1883, 'Zur Geometrie der Alten, insbesondere fiber ein Axiom des Archimedes', Mathematische Annalen, 22, 504-519. Stolz, 0.: 1884, 'Die unendlich kleinen Grdssen', Berichte des NaturwissenschaftlichMedizinischen Vereines in JnnsbrUck, 14, 21-43. Stolz, 0.: 1885, Vorlesungen Uber allgemeine Arithmetik; Erster Theil: Allgemeines und Arithmetik der reelen Zahlen, Teubner, Leipzig. Tarski, A.: 1951, A Decision Method For Elementary Algebra and Geometry, Berkeley and Los Angeles. Thomae, J.: 1870, Abriss einer Theorie der complexen Functionenund der Thetafunctionen einer Veranderlichen, Nebert, Halle. Troelstra, A. S. and van Dalen, D.: 1988, Constructivism in Mathematics I, II, NorthHolland, New York. van Dalen, D. (Ed.): 1981, Brouwer's Cambridge Lectures on Intuitionism, Cambridge University Press, Cambridge. Veblen, 0.: 1903, 'Hilbert's Foundations of Geometry', The Monist, 13, 303-309. Veronese, G.: 1889, '11continuo rettilineo e l'assioma V d'Archimede', Atti Della Reale Accademia Dei Lincei, Memorie (Della Classe Di Scienze Fisiche, Matematiche E Naturali) Roma, 6, 603-624. Veronese, G.: 1891, Fondamenti di Geometria, Padova. Veronese, G.: 1894, Grundziige der Geometrie, A. Schepp, Leipzig. Wallis, J.: 1655, De Sectionibus Conicis, in Opera, 1695, Oxford. Wallis, J.: 1657, Mathesis Universalis, in Opera, 1695, Oxford. Wilson, E. B.: 1904, 'The Foundations of Mathematics' Bulletin of the American Mathematical Society, 11, 74-93.

PART I

THE CANTOR-DEDEKIND PHILOSOPHY AND ITS EARLY RECEPTION

E. W. HOBSON

ON THE INFINITE AND THE INFINITESIMAL IN MATHEMATICAL ANALYSIS PresidentialAddress, by E. W. Hobson, Sc.D., FR.S., November 13th, 1902.*

MR. PRESIDENT,

In the days of our forefathers, when an unsuccessful politician had reached the end of his career, it was customary to grant him one last privilege, that of delivering an address upon topics chosen by himself to the assembled multitude on Tower Hill. Although my conscience acquits me of having been guilty during my period of office of conduct traitorous to the interests of our Society, I avail myself of the corresponding privilege accorded by our custom to a retiring President. The remark that the nineteenth century has been an age of unexampled progress in all branches of science has been so often made as to have become a commonplace. The remark is true in a pre-eminent degree of our own department of science. As is known to you all, at no earlier time has a more rapid development taken place in all parts of mathematical science, involving the creation of entirely new branches and of new and powerful general methods. However, it is not in the main of these new developments and of the extensions made in our science in the outward direction that I propose to speak this evening. In the past century, and perhaps especially during the second half of it, the attention of mathematicians has been devoted in an unusual degree to a critical examination of the foundations of the various branches of mathematical thought. In analysis, geometry, and mechanics a close scrutiny has been made of the fundamental assumptions and concepts. This scrutiny has resulted not only in a large measure of restatement of the base principles of these departments of science, but has also powerfully reacted on the methods of procedure within these departments, and has suggested new and fruitful lines of research. Although outside criticism of the foundations of mathematics has at all times been abundant, the work of underpinning the edifice of our science has been for the most part carried out by the same workmen who have been engaged in the general work of the structure, and especially in building new wings. There are times when it is appropriate to draw attention in general terms to the 3 P. Ehrlich (ed.), Real Numbers, Generalizationsof the Reals, and Theories of Continua, 3-26. © 1994 Kluwer Academic Publishers.Printed in the Netherlands.

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critical side of some part of our science, and I think it will be admitted that a Presidential address is such an occasion. I have accordingly chosen as the subject of my discourse this evening 'The Infinite and the Infinitesimal in Mathematical Analysis.' It will be found that my intention to speak of critical rather than constructional results admits of some considerable exceptions. These exceptions will, however, illustrate the fact that pertinent criticism of fundamentals almost invariably gives rise to new construction. On such a subject as that I have chosen, I cannot hope to have anything essentially new to say; but, nevertheless, I venture to hope it may not be profitless, if I state as explicitly as I can, what seems to me to be the trend of thought in this connexion at present prevailing among mathematicians as the result of the labours of some of the most distinguished of their number during the last half century. I am strengthened in this view by my knowledge of the fact that many British mathematicians, absorbed as they rightly are in the technique of their science, and in the work of applying it to the quantitative description of natural phenomena, have not yet fully appreciated the results of recent movements in mathematical thought in this connexion. In some of the text-books in common use in this country, the symbol - is still used as if it denoted a number, and one in all respects on a par with the finite numbers. The foundations of the integral calculus are treated as if Riemann had never lived and worked. The order in which double limits are taken is treated as immaterial, and in many other respects the critical results of the last century are ignored. It would, however, be unjust not to recognize the fact that a great improvement in these respects has been shown in some of the most recent of our textbooks. Essentially connected as views about the infinite and the infinitesimal are with the most fundamental notions on which analysis is based, with the concepts of number and magnitude, with the notions of continuity and discreteness, with the doctrine of limits in all the various forms in which it has appeared, with the nature of the ideal objects with which mathematical thought operates, these ideas have been a subject of unceasing controversy since the very commencement of abstract thought - a controversy which has by no means ceased at the present time. The fact that these fundamentals lie on the border-line across which mathematics passes into the wider region of philosophy has brought it about that in all ages philosophical thinkers as well as mathematicians, before as well as after the two classes ceased to be identical,

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have occupied themselves with the attempt to introduce clearness into the doctrine concerning them. The kind of judgments which are made in mathematical thinking, forming as they do a class which in certain aspects are of a comparatively simple character, have at all times formed a kind of touchstone on which epistemologists have tested their general theories of knowledge. To attempt to give, even in outline, a history of thought upon the subjects of the infinite and the infinitesimal, involving as it would the task of tracing the history of the various theories of the infinitesimal calculus, would be altogether beyond the scope of such a discourse as the present one. In order, however, to make clear what has been the precise effect of the more recent movements of thought in this order of ideas, it will be necessary for me to take a brief glance at the mode in which the subject presented itself at various times to thinkers confronted with the ordinary problems of mathematical analysis. How, then, did the problems of analysis present themselves to the earliest mathematicians? What were the elements with which those mathematicians had to work? The two notions of number and of magnitude with which they had to operate in problems of a geometrical or kinematical character, have points of resemblance and also points of difference. Both number and magnitude appear by their very nature to be unlimited in two directions: there is no greatest number or magnitude, and (excluding zero) no smallest one. A set of numbers or of magnitudes may be contemplated, each one of which is definite and finite, and yet the set contains numbers or magnitudes which are greater than any particular number or magnitude which we may choose to assign. A similar possibility holds as regards smallness. A symbol to which are assigned successively the increasing or diminishing values of the numbers or magnitudes in the set contemplated, is said to become in the one case indefinitely great, in the other case indefinitely small. At any particular stage the symbol represents a finite number or magnitude, but the absence of a limit is designated by the phrase "becoming indefinitely great, or small." The indefinitely great thus described is the potentially infinite, das uneigentlich Unendliche, and expresses the mere absence of upper limit to a variable. In this form, as expressing a mere potentiality, the infinite and the infinitesimal seem so inevitable a necessity of thought as hardly to give rise to differences of opinion, except, perhaps, upon matters of language. But when it is conceived that these mere potentialities pass into actualities, that fixed numbers

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or magnitudes exist which are infinite or infinitesimal, that the merely indefinitely great becomes an actual infinite, or the merely indefinitely small becomes an actual infinitesimal, the region of serious controversy has been reached - a controversy which is still proceeding, and about the modern aspect of which I shall have some remarks to make later on. In respect of the actually infinite, there have been exhibited at different times and by various thinkers the extremes of faith and of scepticism; there have been believers and sceptics, critics and freethinkers, idealists and empiricists. The infinite of mathematics has at times been treated with that familiarity which is bred of innocent inappreciation. Bold generalizations have been made in which rules applicable to the finite were uncritically and unconditionally extended to the indefinitely great, as if that represented an actuality necessarily subject to the same rules of operation as the finite. At other times, the desire to remain on what was felt to be the firmer ground of empirical knowledge has led almost to a denial of all validity to the conceptions of the infinite and the infinitesimal, and to all processes involving their use. It is noteworthy that both these attitudes of mind have at different times been of direct advantage to science, and that the most opposite tendencies in regard to this order of ideas have led to the advancement of knowledge. One of the principal forms in which an indefinitely great number of operations occurs is that of infinite series, which were introduced in the seventeenth century. The mathematicians of the eighteenth century used these series freely, without troubling themselves much as to questions of convergence. Early in the nineteenth century came a rude awakening. In a letter written by Abel1 in 1826 we read: "Divergent series are in toto an invention of the Devil, and it is a disgrace that any one should venture to found on them the smallest demonstration. One can get out of them anything one likes when one employs them, and it is they which have produced so many difficulties and so many paradoxes." And further: "I have become prodigiously attentive to all that; for, if one excepts the cases of the most extreme simplicity - for example, geometric series - there is scarcely in the whole of mathematics a single infinite series of which the sum is determined in a rigorous manner. In other words, all that is most important in mathematics is without foundation. Most of the things are exact, that is true, and it is extraordinarily surprising. I am trying to find out the reason." In our time, now that the use of divergent series has been to a large extent placed upon a sound

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mathematical basis, the work of Poincar6, Stieltjes, Borel, and others, has given us an answer to the question that puzzled Abel. To the Greeks, and to later thinkers, magnitude, as given by the intuition of space, time, and especially of motion, appeared to present itself essentially as a continuum, the intuitional or sensuous continuum. On the other hand, number (and it must be recollected that the Greeks only knew rational numbers) appeared to be essentially discrete. Fractional numbers arose historically from the necessity for the representation of the sub-divisions of a unit magnitude into equal parts. The Greek discovery of the existence of magnitudes which are incommensurable with a given unit, by exhibiting the inadequacy of such discrete numbers for the complete representation of prima facie continuous magnitude, served to emphasize the distinction contained in the antinomy of the continuous and the discrete. In order completely to envisage the problem of analysis as it presented itself to the minds of mathematicians from the earliest commencement of the attempts to deal with geometrical and kinematical problems numerically, we must take into account that peculiarity of the human mind in virtue of which it is in general unable to deal with an object of thought as a whole, but is obliged to consider it piecemeal, dividing it up into some kind of elements, taking account of these, and reconstructing the object mentally by a process of synthesis. This inability to grasp a scheme of relations at once as a whole, involved as it is in our essentially discursive modes of apprehension, leads to the necessity of dividing up a geometrical figure, or a portion of time, into parts regarded as elements of the whole, dealing separately with these, and of obtaining final judgments as to the integral properties of the figure or the motion, by means of a process of summation. This necessity of mathematical method led directly to a discussion of the nature of the elements of which magnitudes were to be regarded as made up: Could, for example, the straight line be legitimately regarded as made up of points? If so, of how many? Ought it not rather to be regarded as made up of infinitesimal elements, each of which possess all the properties of the finite length? Ought such elements to be regarded as fixed or as essentially in a state of flux? Such were the questions which inevitably presented themselves as soon as men began to investigate geometrically or analytically the properties of curves and surfaces, to determine areas and volumes. Again, in order to deal with a geometrical figure, not only had the figure to be divided up into elements, but qualitative changes in the figure

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had to be introduced: for example, a curve without corners had to be replaced by a rectilineal polygon with corners, if its length was to be found. Here we have the origin of the method of limits, in its geometrical and its arithmetical forms, and here we come across the central difficultly of the mode in which a limit was regarded as being actually attained. A limit, which appeared only as the unattainable end of a process of indefinite regression, and to which unending approach was made, had, by some process inaccessible to the sensuous imagination, to be regarded as actually reached; the chasm which separated the limit from the approaching magnitudes had in some mysterious way to be leapt over: the attainment of a numerical limit, and an actual qualitative change in a geometrical figure were to be regarded as somehow taking place simultaneously as the result of a process which contained no principle of termination within itself. The germ of the methods of the infinitesimal calculus appears in a geometrical form in the method of exhaustions, employed by the Greek geometers. It was by this method that Archimedes showed that the area of the surface of a sphere is four times that of a great circle, by which he expressed the area of the surface of a right cone, and solved other problems of a similar nature. We have an example of this method in the proof in Euclid XII. 2, that the circumferences of circles are as their diameters; in this case the quantity to be determined is trapped in between two sets of polygons, the one circumscribed to, and the other inscribed in, the circle; as the number of sides of the polygons is increased the space between the two sets of polygons is exhausted; the proof that the required result is obtained is then carried out by the method of reduction ad absurdum. This method would, in the cases in which it can be carried out, leave nothing to be desired as regards rigour, provided the existence of the limit is a priori admitted. It will be observed that the Greeks did not deem it necessary to define the length of a circle, or other curve; that every curve has a length, and every surface an area, was taken by them to be a truth obvious from intuition. Naturally they were not led by intuition to contemplate the existence of not-rectifiable curves. In the method of indivisibles employed by Cavallieri, Pascal, Roberval, and others, straight lines are regarded as made up of an infinite number of points, surfaces as made up of lines, and volumes of surfaces. This method was applied with considerable success before the introduction by Newton and Leibniz, of the methods of the Infinitesimal Calculus, but

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it appears to have been regarded, by some at least of those who employed it, in the light of a shortened mode of procedure in which the method of exhuastions is used with an abbreviated form of language, rather than as a method, the principles of which, when taken literally, were to be regarded as rigorous. Thus Pascal writes: "J'ai voulu faire cet avertissement, pour montrer que tout ce qui est d6montrd par les v6ritables r~gles des indivisibles, se d6montrera aussi ý la rigueur et A la mani~re des anciens; et qu'ainsi l'une de ces mdthodes ne diff~re de l'autre qu'en la mani~re le parler; ce qui ne peut blesser les personnes raisonnables quand on les a une fois averties de ce qu'on entend par The infinitesimal calculus, in the form which was devised by Leibniz, has usually been regarded as the art of employing infinitesimal quantities as auxiliaries for the purpose of finding the relations between certain quantities of which the existence is assumed. In order to find the relations between certain quantities, some of which are constant, others variable, the system is imagined as having arrived at a determinate state regarded as fixed; this state is then compared with other states of the same system, which are regarded as continually approaching the first state, so as to differ arbitrarily little from it. These other states of the system are regarded only as auxiliary systems introduced to facilitate the comparison between the parts of the fixed one. The differences of corresponding quantities in all these systems can be regarded as arbitrarily small, without changing the quantities which define the fixed state, and the relations between which are to be found; these differences are the infinitesimals, and unity divided by one of these infinitesimals was regarded as giving rise to an infinite quantity. The question as to the true nature of these infinitesimals gave rise to almost endless discussions; the views which have been maintained with regard to them fall under three main heads. By some, the infinitesimals have been regarded as fixed objects, having a real existence and in a state of rest outside the ordinary realm of magnitude, two finite magnitudes which differ by an infinitesimal being regarded as equal to one another. A second view as regards infinitesimals is that they are ordinary magnitudes essentially in a state of motion towards zero. This conception of magnitudes continually in a state of flux has been sarcastically described by P. Du Bois-Reymond as follows: - "As long as the book is closed there is perfect repose, but as soon as I open it there commences a race of all the magnitudes which are provided with the letter d towards

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the zero limit." The third view as to the nature of the infinitesimals is that they jare simply ordinary magnitudes too small to be perceived by the senses, and possessing thus only a relative smallness. This view is that of those mathematicians who regard a geometrical point as simply an object whose size is too small to be perceived by the means at our command; a line as a volume of which two of the dimensions are insensible; and so on. The empiricists of this school refuse to idealize objects of perception which form the subject of calculations, by bringing them under exact abstract definitions; the calculus thus regarded is an approximative system in which the results make no claim to absolute exactness, but only to freedom from errors which are observable. Apart altogether from the difficulties as to the true nature of the differentials, it will be observed that in the Liebnizian calculus the existence of the magnitudes between which the relations are in any special problem to be found is regarded as a priori known, or, in other words, no doubt is admitted as to the existence of the limit; this it has in common with the Greek method of exhaustions, of which it is essentially a translation into a more analytical and convenient form. In the method of limits devised by Newton, and employed in a different form by later writers, infinitesimals are not employed singly, but the ratio of two quantities at the moment when they vanish is contemplated, and forms what was later known as the differential coefficient. These "ghosts of departed quantities," as Bishop Berkeley derisively designated them, whose ratio at the moment of their disappearance is the quantity dealt with, present very much the same kind of difficulty as in the Leibnizian form of the calculus. No criterion was obtained for the determinacy of such an ultimate ratio, whose existence was regarded as obvious from intuition. In that form of the Newtonian calculus known as the method of fluxions, the appeal to intuition was made more cogent by representing the vanishing ratio in the form of a velocity; that a moving point has at every instant necessarily a definite velocity was apparently hardly doubted until comparatively recently. We now know that such a velocity has no such unconditional existence as was supposed. Speaking of the method of vanishing ratios, Lagrange writes "Cette mdthode a le grand inconvenient de consider les quantities, dans l'6tat oil elles cessent, pour ainsi dire, d'8tre quantit6s; car quoiqu'on conqoive toujours bien le rapport de deux quantit6s, tant qu'elles demeurent finies, ce rapport n'offre plus t l'esprit une idWe claire et precise, aussit6t que

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ces termes deviennent l'un et l'autre nuls A la fois." This clear perception on the part of Lagrange of the difficulty at the root of the method of limits or of differential coefficients, was doubtless a determining factor in deciding him to embark upon his great attempt to place the calculus upon a basis independent of the idea of infinitesimals or of their ratios. The title of his great work, "Th6orie des fonctions analytiques, contenant les principes du calcul diff6rentiel, d6gag6s de toute consideration d'infiniment petits, d'6vanouissans, de limites et de fluxions, et r6duits A l'analyse algdbrique des quantit6s finies," contains the most concise statement of his aim. Although his attempt was in principle a failure, his idea of making Taylor's series the cardinal form by which functions are to be represented must be regarded as containing the germ of the theory of analytical functions which was developed with so much success at a later period. In the various forms of the infinitesimal calculus to which I have referred, a crucial difficulty is that of the existence of the limit. That this difficulty is no merely imaginary one, but indicates a real gap in the logical basis of the systems, receives an a posteriori confirmation from the discoveries made in the latter half of the nineteenth century, that special restrictions in the nature of the functions employed are necessary for the validity of the ordinary processes of the calculus. The exhibition by Weierstrass and others, of continuous non-differentiable functions, the resulting investigations of the restrictive conditions over and above that of continuity which are necessary for the existence of a differential coefficient, Riemann's investigation of the conditions of integrability of a function, the various theorems discovered as to the conditions of the reversibility of the order of double limits, all indicate that the existence of a limit cannot be presumed apart from all restrictive conditions. The failure of the older analysis to exhibit the existence and nature of such restrictive conditions is a clear proof of defectiveness in the logical basis of that analysis. That the earlier mathematicians were usually able to obtain correct results by means of their methods, is due to the fact that the functions with which they operated were of a comparatively simple character; in point of fact, almost all the functions which are required for the investigation of the problems arising from ordinary intuition satisfy the restrictive conditions first brought to light in our day. I now come to consider the changes which have been brought about in the point of view of mathematicians with respect to the matters I

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have discussed, as the result of the critical efforts of recent times. In the first place, the notion of number, integral or fractional, has been placed upon a basis entirely independent of measurable magnitude, and pure analysis is regarded as a scheme which deals with number only, and has, per se, no concern with measurable quantity. Analysis thus placed upon an arithmetical basis is characterized by the rejection of all appeals to our special intuitions of space, time, and motion,2 in support of the possibility of its operations. It is a very significant fact that the operation of counting, in connexion with which numbers, integral and fractional, have their origin, is the one, and only absolutely exact, operation of a mathematical character which we are able to undertake upon the objects which we perceive; this is due to the fact that the operation is of a highly abstract character, since in counting objects, all special qualitative or quantitative peculiarities of the objects counted are treated as irrelevant. On the other hand, all operations of the nature of measurement which we can perform in connexion with the objects of perception contain an essential element of inexactness, corresponding to the approximative character of our sensuous intuition. The theory of exact measurement in the domain of the ideal objects of abstract geometry is not immediately derivable from intuition, but is now usually regarded as requiring for its development a previous independent investigation of the nature and relations of number. The relations of number having been developed on an independent basis, the scheme is applied by the help of the principle of congruency, or other equivalent principle, to the representation of extensive or intensive magnitude. In any such theory of measurement the non-arithmetical conception of a unit is involved. Those departments of science, including geometry, in which abstract measurement is applied are thus regarded as fields of application for analysis; but they do not directly contribute towards the development of pure analysis, although they may, no doubt, suggest to it problems for treatment in accordance with its own principles. This complete separation of the notion of number, especially fractional number, from that of magnitude, involves, no doubt, a reversal of the historical and psychological orders. It is, however, no uncommon occurrence that the logical order of a subject should be very different from the historical order in which the concepts of the subject have arisen. Is it not an essential part of our scientific procedure that, in our conceptual schemes, factors are separated from one another, which intuitionally appeared in combination?

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The so-called arithmetization of analysis is, and has been, accepted in somewhat various degrees by different mathematicians. The extreme arithmetizing school, of which, perhaps, Kronecker was the founder, ascribes reality, whatever that may mean, to integral numbers only, and regards fractional numbers as possessing only a derivative character, and as being introduced only for convenience of notation. The ideal of this school is that every theorem of analysis should be interpretable as giving a relation between integral numbers only. The validity and feasibility of this ideal I cannot here discuss. Some mathematicians, on the other hand, like P. Du Bois-Reymond, while using to a large extent the ideas and methods of arithmetical analysis, appear still to regard the notion of continuous magnitude as a necessary part of the foundations of the subject. The true ground of the difficulties of the older analysis as regards the existence of limits, and in relation to the application to measurable quantity, lies in its inadequate conception of the domain of number, in accordance with which the only numbers really defined were rational numbers. This inadequacy has now been removed by means of a purely arithmetical definition of irrational numbers, by means of which the continuum of real numbers has been set up as the domain of the independent variable in ordinary analysis. This definition has been given in the main in three forms - one by Heine and Cantor, the second by Dedekind, and the third by Weierstrass. Of these the first two are the simplest for working purposes, and are essentially equivalent to one another; the difference between them is that, whilst Dedekind defines an irrational number by means of a section of all the rational numbers, in the Heine-Cantor form of definition a selected convergent aggregate of such numbers is employed. The essential change introduced by this definition of irrational numbers is that, for the scheme of rational numbers, a new scheme of numbers is substituted, in which each number, rational or irrational, is defined and can be exhibited in an indefinitely great number of ways, by means of a convergent aggregate of rational numbers. In this continuum of real numbers the notion of number is, as it were, raised to a different plane. By this conception of the domain of number the root difficulty of the older analysis as to the existence of a limit is turned, each number of the continuum being really defined in such a way that it itself exhibits the limit of certain classes of convergent sequences. It would, of course, be futile to define a number by means of a convergent aggregate, were it not shown - as has been,

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in fact, done - that the ordinary operations of arithmetic can be defined for such numbers in such a way as to be in agreement with the ordinary scheme of operations for the rational numbers taken on the lower plane. It should be observed that the criterion for the convergence of an aggregate is of such a character that no use is made in it of infinitesimals, definite finite numbers alone being used in the tests. The old attempts to prove the existence of limits of convergent aggregates were, in default of a previous arithmetical definition of irrational number, doomed to inevitable failure. It could not, for example, in general be shown that an unending decimal formed according to prescribed rules possessed a limit, since it was clearly impossible to infer from the existence and properties of a set of rational numbers, the existence of a number which itself is in general not rational, and was therefore undefined within the domain of operation. A considerable part of the newer analysis consists of putting the criterion for the convergence of an aggregate into various forms suitable for application in various classes of cases. The convergence of an aggregate having in any given case been established by the application of one or other of such derivative rules, the aggregate itself defines the limit. In all such proofs the only statements made are as to relations of finite numbers, no such entities as infinitesimals being recognized or employed. Such is the essence of the £ proofs with which we are familiar. In such applications of analysis - as, for example, the rectification of a curve - the length of the curve is defined by the aggregate formed by the lengths of a proper sequence of inscribed polygons. The length is not regarded as something whose existence is a priori known. In case the aggregate is not convergent, the curve is regarded as not rectifiable. If it can be shown that the lengths of these inscribed polygons form a convergent aggregate which is independent of the particular choice of the polygons of the sequence, the curve is rectifiable, its length being defined by the number given by the aggregate. The older analysts regarded the domain of the real variable, or of a set of real variables, as the continuum given by our intuition of space, time, and motion; this continuum was usually accepted uncritically as a notion completely given by intuition and hardly capable of further analysis; however, those points of the continuum which could be represented by number (rational number) formed only a discrete aggregate, and thus the variable had to pass through values which were not definable as numbers. This intuitive notion of the continuum appears

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to have as its content the notion of unlimited divisibility, the facts that, for instance, in the linear continuum we can within any interval PQ find a smaller one P'Q', that this process may be continued as far as the limits of our perception allow, and that we are unable to conceive that even beyond the limits of our perception the process of divisibility in thought can come to an end. However, the modern discussions as to the nature of the arithmetic continuum have made it clear that this property of unlimited divisibility, or connexity, is only one of the distinguishing characteristics of the continuum, and is insufficient to mark it off from other domains which have the like property. The aggregate of rational numbers, or of points on a straight line corresponding to such numbers, possesses this property of connexity in common with the continuum, and yet is not continuous; between any two rational numbers another pair can be found, and this process may be continued until we obtain an arbitrarily small interval. The other property of an aggregate which is characteristic of a continuum, is that of being, in the technical language of the theory of aggregates (Mengenlehre) perfect: the meaning of this is that all the limits of convergent sequences of numbers or points belonging to the aggregate themselves belong to the aggregate; and, conversely, that every number or point of the aggregate can be exhibited as the limit of such a sequence. The aggregate of rational numbers does not possess this property of being perfect, since the limit of a sequence of such numbers does not necessarily belong to the aggregate. That the aggregate of rational numbers is not perfect, or even closed, is the root defect of that aggregate, which led to the difficulty as regards the existence of limits, in the older analysis. The two properties of connexity and of perfection are regarded as the necessary and sufficient characteristics of a continuum; it is remarkable that in analysis the latter property of a continuum, which was not brought to light by those who took the intuitive continuum as a sufficient basis, is in some respects the more absolutely essential property for the domain of a function which is to be submitted to the operations of the calculus. It has in fact been shown that many of the properties of functions, such as continuity, differentiability, are capable of precise definition when the domain of the variable is not a continuum, provided, however, that domain is perfect; this has appeared clearly in the course of recent investigation of the properties of non-dense perfect aggregates, and of functions of a variable whose domain is such an aggregate. The arithmetical continuum having been defined and explored, it is

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then postulated that on a straight line there exists one point, and one only, corresportding to each number of the arithmetical continuum, and that no other points exist on the straight line; this fixing of the point contents of a straight line amounts to an exclusion of the contemplation of fixed infinitesimal lengths. Similarly, it is postulated that in three-dimensional space there exists one point, and one only, corresponding to each specification of three coordinates of the point by means of numbers, and that the points whose existence is thus postulated exhaust the space. The arithmetizing school thus regard the nature of the geometrical continuum as being cleared up and described by means of the previously defined arithmetical continuum; this, is of course, a reversal of the traditional view. The view I have sketched of the philosophy of the continuum does not meet with the universal acceptance of mathematicians, as an adequate scheme, at the present time. As an example of a rival scheme I may briefly touch upon the one propounded by Veronese. He develops the notion of the abstract linear continuum from the intuitive side, and traces the consequence of supposing that on a straight line two intervals PQ, P'Q' can co-exist such that the smaller P'Q' is so small compared with PQ, that no integer n can be found which will make n. P'Q' exceed PQ, thus rejecting what is known as the axiom of Archimedes. This amounts to the affirmation of the existence of fixed infinitesimal lengths, and of fixed infinite lengths, on the straight line. In this scheme, when a unit length is chosen on the straight line, Dedekind's section of rational points is made, not by a single point, as in the Cantor-Dedekind scheme, but by an infinitesimal length, that is, by a length which is infinitesimal relative to the scale of measurement chosen. Veronese contemplates the existence of an indefinite series of scales in the linear continuum, such that each unit is infinite compared with one belonging to a lower scale, and is infinitesimal compared with a unit belonging to a higher scale; he then proceeds to introduce a scheme of infinite and infinitesimal numbers which will suffice for the complete representation of points of the straight line. On this view, the DedekindCantor continuum, when represented on the straight line, is only a relative continuum, that is, relative to the particular scale employed in the representation; the absolute continuum would require for its representation an indefinite series of infinite and infinitesimal numbers. As to the validity of Veronese's scheme, that is, as to its consistency with a logical theory of magnitude, I do not propose to express any opinion;

ON THE INFINITE AND THE INFINITESIMAL

17

the matter has been a subject of considerable controversy. Assuming, however, its validity as a possible scheme, it does not affect the validity of the Cantor-Dedekind scheme; the comparative simplicity of the latter would indicate it as the natural basis for analysis, and for the applications to the measurement of magnitude. One of the most interesting results on the speculative side of abstract science, which has been obtained in the nineteenth century, is that it is possible to set up two or more conceptual systems, each self-consistent, but contradictory with one another, each of which provides a sufficient representation of the facts of perception; the most striking example of this has been in geometry, where it has been shown that, under a certain limitation, Euclidean, hyperbolic, and elliptic geometry may each afford a sufficient representation of the properties of figures in perceptual space. We are entitled to postulate the existence of whatever points we choose upon that ideal object, the line of geometry, provided our scheme does not contradict itself, and, further, provided the ideal object thus constituted affords an adequate representation of the concrete lines which we perceive in the external world. Between two such schemes intuition can make no choice, and in abstract science we make that choice between them which is dictated by considerations of simplicity and of suitability for the special purpose on hand. The question as to the legitimacy of the use of infinite numbers, that is, not merely of the use of a variable which is regarded as becoming indefinitely great, but of numbers which are actually infinite and to be regarded as capable of entering into relations, is a matter which has been discussed by philosophical thinkers from the time of Aristotle onwards. The balance of opinion seems to have been decidedly against the validity of the conception of such numbers; in support of this negative view, Aristotle himself, Locke, Descartes, Spinoza, and Leibniz may be quoted. The grounds of the objection to the introduction of such numbers may in the main be reduced to three heads. First, it is said that a number is, by its very nature, finite: this is supported by the plea that all actual operations of counting and measuring are performed upon finite aggregates or finite magnitudes; to refute this view, it may be urged that the introduction of infinite numbers, if it can be made at all, will justify itself by a proof of the capability of such numbers for the representation and characterization of non-finite aggregates; in fact, it may be held that the objection contains a petition principii. Secondly, it has been widely held that a scheme of infinite numbers represents an

18

E. W. HOBSON

endeavour to make distinctions and determinations within the infinite; whereas the true infinite admits of no determination. If the infinite be identified with the all-embracing absolute of idealistic philosophy, it will probably be admitted that such an absolute admits of no distinctions, for "omnis determinatio est negatio"; however, the question arises whether a domain may not exist which, though not finite, is still not to be regarded as engulfed in the absolute, and which therefore may still in some measure admit of definition and determination, and which may require a special non-finite system of number for the specification of its characteristics; such an intermediate domain has been named by Cantor the "transfinite" or "superfinite." Thirdly, it has been urged that finite numbers would be unable to maintain themselves as against infinite ones; that the finite and its relations would be absorbed in the infinite, and could enter into no relations with it: the value of this objection can be estimated a posteriori only, if and when a system of infinite or transfinite numbers has been actually defined and the nature of its connexion with the finite brought to light. That mathematicians still shrink from leaving what they regard as the firm ground of the finite based upon experience is illustrated by a remark in an introductory passage in Tannery's work: Introduction ti la Thdorie des Fonctionsd'une Variable rielle.He writes, "On peat constituer enti~rement 1'analyse avec la notion de nombre entier et les notions relatives ý addition des nombres entiers; il est inutile de faire appel a aucun autre postulat, ý aucune autre donn6e de l'exp6rience, la notion de l'infini dont il ne faut pas faire myst~re en mathdmatique se r6duit i ceci, apr~s chaque nombre entier il y'a un autre." However sufficient the restriction to the merely indefinitely great, here indicated, may be for the more ordinary purposes of analysis, provided, however, that an exploration of the properties of the continuum of real numbers is not carried too far, I hope to be able to show, as clearly as possible in the brief space at my disposal, that the introduction by Cantor of systems of transfinite numbers is justified by the primary necessities of our analytical system; it may be justified in point of utility by the numerous applications which are being made, both in analysis and in geometry, of the conceptions and results of the theory of aggregates, to express the characteristics of which these transfinite numbers are required. No mathematician will wish to make a mystery of the infinite in analysis; mathematics has nothing to do with mysteries except to endeavour to remove them. It is to be remarked that the introduction into analysis of the transfinite numbers

ON THE INFINITE AND THE INFINITESIMAL

19

was historically by no means the result of a purely speculative tendency to explore the unknown and mysterious, and certainly did not arise from any taste on the part of their inventor for "tricks to show the stretch of human brain"; their introduction arose principally out of the necessities of investigations connected with the peculiarities of Fourier's series and of the functions representable by such series. Cantor writes: "Zu dem Gedanken, das Unendlichgrosse nicht bloss in der Form des unbegrenzt Wachsenden und in der hiermit eng zusammenhdngenden Form der im siebenzehnten Yahrhundert zuerst eingefUhrten convergenten unendlichen Reilhen zu betrachten, sondern es auch in der bestimmten Form des Vollendetunendlichen mathematisch durch Zahlen zu fixiren, bin ich fast wider meinen Willen, weil im Gegensatz zu mir werthgewordenen Traditionen, durch den Verlauf vieljdhriger wissenschafticher Bemtihungen und Versuche logisch gezwungen worden." The first real breach in the infinite - one which established a true line of cleavage - was made when Cantor showed that the aggregate of rational numbers is enumerable, whereas the aggregate of real numbers, rational and irrational, is unenumerable. This denotes that a (1, 1) correspondence can be established between the rational numbers in any given interval, and the aggregate of positive integers, whereas no such correspondence can be established between the numbers of a continuum and the aggregate of integral numbers. Thus the rational numbers can be counted and the irrational numbers cannot be counted. All the rational numbers in any interval can be arranged in a definite order (not of magnitude), so that one of them stands first and each particular number has its assigned place. No such arrangement can be made when the irrational numbers of the interval are taken into account. This far-reaching result brings out in a strong light the difficult nature of the conception of the continuum as a given totality. If it be asked in what sense can the numbers of the continuum be considered as forming a given or determine aggregate, we must contrast this aggregate with that of the rational numbers or with that of the integral numbers. These latter are not, of course, given in the sense that we can exhaustively exhibit them by means of symbols on a sheet of paper. We could only do that in the case of a finite aggregate; but they are given in the sense that we can say of any particular number where it is to be found in a regularly arranged scheme. On the other hand, the aggregate of all real numbers is not given in the same sense; no rule, and no set of rules, can be given by which we could obtain successively all the numbers

20

E. W. HOBSON

of the aggregate, so that each particular number would necessarily appear in the course of the procedure; and this is a consequence of the unenumerable character of the aggregate. The aggregate of real numbers can be regarded as given only in the sense that every possible real number that we may choose to define by means of an analytical process belongs to the aggregate. This somewhat negative conception of its determinacy is an essential characteristic of the unenumerable aggregate. How far the mathematicians of the future will rest satisfied with this conception of the arithmetic continuum, time alone can decide. When we count a finite number of objects we take them in some definite order, and establish a correspondence between them and the ordinal numbers. The last ordinal number employed, we call the ordinal number, or simply the number of the collection. When we take into account the fundamental property that this number is independent of the order in which the objects are counted, we identify this number with the cardinal number of the collection. Thus, in dealing with finite aggregates, the distinction between the ordinal and the cardinal number, though logically existent, may be practically disregarded. This, however, is no longer the case when we deal with non-finite aggregates; here cardinal and ordinal numbers must be kept quite distinct, and their properties must be developed on different lines. The theorem that the ordinal number of an aggregate is unaltered by changing the order of the elements of the aggregate no longer holds. In order to exhibit the way in which transfinite ordinal numbers are required when we deal with non-finite aggregates, I propose to refer to a well known paradox, that of Achilles and the tortoise, which in various forms has afforded an interesting exercise to logicians. Achilles goes ten times as fast as the tortoise, and the latter has ten feet start. When Achilles has gone ten feet the tortoise is one foot in front of him; when Achilles has gone one foot further the tortoise 1/10 ft. in front; when Achilles has gone 1/10 ft. further the tortoise is 1/100 ft. in front; and so on, without end; therefore Achilles will never catch the tortoise. The fallacy, of course, lies in the surreptitious transcending of the convergent process when the word "never" is used in the conclusion. Let us indicate the successive positions of Achilles referred to, by the ordinal numbers 1, 2, 3, . . . suffixed to the letter A, so that A 1, A 2, A 3 . . .. represent the positions of Achilles mentioned in the paradox. These points A1, A 2, A 3.... have a limiting point, which represents the place

ON THE INFINITE AND THE INFINITESIMAL B2

B 3B 4 B.

A,

A2A3A4A.o

21

A(0+xAwo+ 2Ao 2

where Achilles actually catches the tortoise. This limiting point is not contained in the set of points Al, A2, A 3, . . . ; if we wish to represent it, we must introduce a new symbol co, and denote the point by Aw,. This symbol (o represents Cantor's first transfinite ordinal number. It does

not occur in the series 1, 2, 3 ....

but is preceded by all these numbers,

and yet there is no number immediately preceding it; it is the first of a

new series of numbers. I may now, perhaps, be allowed to tamper with this classical paradox

to the extent of supposing that there is a second tortoise moving at the same rate as the first, and ten feet in front of it, and of supposing that we wish to represent the positions of Achilles when he is 1 ft., 1/10 ft., 1/100 ft ..... behind the second tortoise. To represent these positions we naturally take A.,,, A.,+ ; the place where Achilles 2 , A.3 .... catches the second tortoise will be denoted by A.,+0, or A112. These numbers, to + 1, co + 2, (o+ 3.... ,o2, are the transfinite ordinal numbers immediately following (o.It thus appears that, as (o + 1 succeeds to, the two cannot be regarded as identical; thus, (o+ 1 > o. If now we had commenced by denoting the positions of Achilles when he was 100, 10, 1, 1/10, 1/100 ....

ft. behind the first tortoise by B 1, B 2,

B 3,

.... so that A. and B,,, represent the same point, we see that the place where the tortoise is caught would still have to be represented by B.,; it is, consequently, necessary to distinguish between co + 1 and 1 + (o, and to write 1 + Co = (o. The two relations Co= 1 + (o, (o + 1 > (o, enable us to illustrate the extent to which a finite number is able to maintain itself against a transfinite one. The above perennially instructive example of a limiting process at once throws light upon the relation of the unending process to the limit, and upon the necessity for the introduction of transfinite numbers for the representation of the limit which is not itself contained within the region of the convergent process. If we could imagine that we had no independent knowledge of the position and existence of the point at which Achilles overtakes the tortoise, we should be in the position of the older analyst or geometer in face of most of the problems which he solved. In the above paradox we have, by artificially involving ourselves in a convergent process, placed the limiting point, and all points beyond it, outside the reach of the process itself; for the representation of this

22

E. W. HOBSON

point and the points beyond it we have to commence anew with a fresh series of ordinal numbers. The only reason why, in ordinary life, transfinite numbers are unnecessary is because we do not make use of such convergent processes. It would be easy for me to arrange artificially a series of points of time during the delivery of the present address which would be such that the moment of the termination of my address could only be represented by a transfinite number; higher transfinite numbers would be required to denote the times of all subsequent events this evening. Although, for the reason I have indicated, transfinite numbers are unnecessary for the purposes of ordinary life, this is by no means the case in certain departments of mathematical analysis, where we are in many cases compelled to make use of convergent processes. It appears that a region which from one point of view belongs to the finite may from another point of view belong to the transfinite, and it is frequently just this latter point of view which the exigencies of analysis compel us to adopt; hence arises the necessity for, and the justification of, the use of transfinite ordinal numbers. I propose, as an illustration of the use of transfinite ordinal numbers, to give a simple means which I have devised for their systematic representation by a set of points on a given finite segment of a straight line. On the straight line AB let us denote by P0 , P1 , P2, P 3, .-. . those points at which the expression logk(AB/PB), where k is a fixed number greater than unity, has the values 0, 1, 2, 3 ... Po A

P.

P1P 2P3

B

the point P 0 coincides with A, and the point B can only be represented by P.. Now take any one of the segments PPr.; this we may for convenience represent on an enlarged scale; denote by Q, 0 , Qri, Q, 2 , ...the points on P•rPrl, at which log1(PrPr,1/QPr, 1) takes the values 0, 1,2,3, ... ; aQar QQr. Pr

............ Pr+l

thus Pr~i can only be represented by Qro. Suppose this to have been done with every segment of AB, and now imagine all the points Q to be marked on AB, and to be numbered from left to right;

ON THE INFINITE AND THE INFINITESIMAL

in POP, we shall have in P1P2 " in P 2P 3

23

0, 1, 2, 3, .... 03, 0)+ 1, 0 + 2, ..... 02, 0)2 + 1, 0)2 + 2 ..... 0,3;

the point B can be represented by 0ox or 0)2. If now we proceed to take each segment QQ,Q.. 1 , and to divide this in a similar manner at points R for which logk[(QrsQr.s+l)/(RQr,s+l)]has the values 0, 1, 2, 3 ..... and then imagine all the points R thus obtained to be marked on the original straight line AB, and numbered as before, from left to right, it will be seen that all the numbers 0)2p + 0)q + r will be required, and that the point B can be represented by 0)33. The points P0 , P1. . .. P, 0... )03;the point Qrs will have for their ordinal numbers 0, 0)2, 0)22 . 2 will be numbered 0O r + 0os; the finite numbers are all used up in the first sub-segment of AB. By proceeding in a similar manner to further sub-division, we may exhibit on AB the ordinal numbers Con"p+ O3n-'p n_1+ . - -+ P O, and the point B will then be represented by 0con+. The distance from A of the point represented by any of these numbers can be easily expressed. I turn now to the subject of the transfinite cardinal numbers. Every two finite aggregates between the elements of which a (1, 1) correspondence can be established have the same cardinal number, and the cardinal number of an aggregate is independent of the order in which the objects are arranged. The extension of this notion of cardinal number to non-finite aggregates leads to the conception of the power (Mdchtigkeit, puissance) of an infinite aggregate, which power is represented by a transfinite cardinal number. Two infinite aggregates between the elements of which a (1, 1) correspondence can be established have the same power or cardinal number. Thus, the aggregate of all rational numbers, or of rational numbers in a given interval, has the same power or cardinal number as the aggregate of integral numbers. This may be denoted by a, and is the first transfinite cardinal number. As I have before mentioned, the continuum of real numbers cannot be placed in (1, 1) correspondence with the integral numbers, and has, therefore, a power or cardinal number different from a; this is usually denoted by c. It is a surprising fact that a continuum of two, three, or any number of dimensions has the same power c as that of the linear continuum i.e., a (1, 1) correspondence exists between all points in an area or in a volume and the points in a straight line, or in a finite segment of a straight

24

E. W. HOBSON

line. It was, in fact, until recently supposed that all known aggregates have either the power of the aggregate of natural numbers or else that of the continuum. It has, however, now been shown that the aggregate of all possible functions of a variable x of which the domain is a continuous interval (a, b) has a power higher than that of the continuum; this higher cardinal number is denoted by f. It should, however, be observed that, if the function is restricted to be analytic, the aggregate of such functions then has the power of c of the continuum. The numerous attempts which have been made to prove that a and c are consecutive cardinal numbers - that is to say, that no aggregate exists whose power exceeds a and is less than c - have hitherto been unsuccessful. This remains, for the present, as a hiatus in the theory of transfinite cardinal numbers. I have here been able to touch only the fringe of the subject of transfinite numbers; but my object has been to indicate, as clearly as is possible in the necessarily brief space I have allotted to them, how they necessarily arise when we try to investigate the peculiarities of non-finite aggregates. The profound study of these numbers and their relations, and especially of their connexion with the theory of the types of ordered aggregates, which has been made by G. Cantor, who, I am proud to remember, has been added to our list of foreign members during my term of office, has resulted in the creation of a veritable arithmetic of the infinite, which seems to be destined to have an ever-increasing range of application in analysis and geometry. The place which the conception of the infinite occupies in the various schemes of geometry, especially the manner in which infinite elements are adjoined to the finite elements, is a subject on which many interesting remarks might be made. This, however, does not belong to the subject of my discourse this evening, and would, in any case, better be left for treatment by some one more competent to discuss it than I can claim to be. In the minds of many men who are engaged in the active work of assisting in the progressive development of science, there is a certain impatience with what they are apt to regard as a hypercritical attitude towards fundamental concepts; this feeling of impatience exists perhaps in exceptionally large measure in the English mind, whose genius is in a preponderating degree directed towards the concrete, and upon which, to a considerable extent, purely abstract questions seem to exercise a peculiar repulsion. However, taken on the whole, the impulse towards clear thinking, which leads men to make an ever renewed dissection of

ON THE INFINITE AND THE INFINITESIMAL

25

the fundamentals of science, and to an ever renewed attempt to state fundamental principles in a form which shall satisfy more nearly the canons of logical thought, is an ineradicable tendency of the human mind, and I, for one, cannot but regard its presence as one of the conditions which are in the long run necessary to render possible the progress of scientific knowledge. Even from the point of view of those who regard mathematics as existing exclusively for the purposes of physical research, it is a short-sighted policy to discourage that free development of mathematics on its abstract side, which is probably a necessary stage in the process of sharpening the tools which mathematics provides for the use of the physicist. Even in the remarks on one aspect of our science which I have made this evening, I think it has been apparent that criticism, and even erroneous criticism, is not infrequently the parent of construction. It may well be true that perfect intellectual transparency with regard to the fundamentals of any branch of knowledge is an unattainable ideal. May it not even be the case that a perfect comprehension of anything would involve a perfect comprehension of everything? Are there not those with us who assert that an analysis of the abstract creations of the human mind inevitably, when pushed far enough, leads to contradiction, and that this is a necessary consequence of the divorce of these ideal objects from reality? However this may be, the thinkers of each age must do what they can with the possibilities open to them, in faith that, however far short of what is completely satisfactory to the intellect may be the results to which they are led, the efforts of their generation in the direction of criticism; as well as of construction, may be found by those who come after them not to have been entirely fruitless. NOTES , This address first appeared in the Proceedings of the London Mathematical Society

35, 117-140 (1902). We are grateful to the LMS for permission to publish it here.

SSee

Abel's correspondence, p. 16, in the volume Niels Henrik Abel: Mimorial public a l'occasion du centenaire de sa naissance.

It is not intended here to prejudge the questions as to the part which intuition may have in the formation of the concepts of number. 2

26

E. W. HOBSON APPENDIX TO PRESIDENT'S ADDRESS

A few references to the literature of the subject are added. IrrationalNumbers and Limits Dedekind, Stetigkeit und irrationaleZahlen, Brunswick, 1872 (English translation in Dedekind, Essays on Number, Chicago, 1901). Heine, Die Elemente der Functionenlehre, Crelle, J. f Math., Bd. LXXXIV, 1872. Tannery, J., Thgorie des Fonctions d'une Variable, Paris, 1886. Pringsheim, 'Irrationalzahlen und Konvergenz unendlicher Prozesse,' in Ency. d. math. Wiss., Bd. I, Heft 1, Leipzig, 1898. Mathews, 'Number,' in Ency. Brit. Supplement, Vol. XXXI, Edinburgh, 1902. Love, 'Functions of Real Variables,' in Ency. Brit. Supplement, Vol. XXVlII. Arithmetized Analysis Dini-Luroth, Grundlagenfar eine Theorie der Functionen einer verdderlichen reellen Grbsse, Leipzig, 1892. Jordan, Cours d'Analyse, Paris, 1893-1896. Klein, Anwendung der Differential- une Integralrechnung auf Geometrie, Leipzig, 1902. For Kronecker's complete arithmetization referred to on p. 13, see his memoirs in Crelle, J. f. Math., Bde. XCII., CI., 1882, 1887. For the theory of divergent series referred to on p. 7, see Borel, Legons sur les Sdries divergentes, Paris, 1901. For philosophical discussions of the foundations of analysis, see P. du Bois Reymond, Allegemeine Functionenlehre, Ttibingen, 1882 (French translation by Milhaud and Girot, Nice, 1887); Couturat, De l'Infini mathdmatique, Paris, 1896; Poincard, La Science et l'Hypoth~se, Paris [19021. Theory of Aggregates and Transfinite Numbers Cantor's chief memoirs are in Acta Mathematica, tt. II., VII., and Math. Ann., Bde. XVII., XX., XXI., XXIII., XLVI., XLIX. A sketch of the theory is given by Schoenflies in Ency. d. math. Wiss., Bd. I., Heft 2, Leipzig, 1899, and a more complete account by the same writer in Jahresberichted. Deutschen Math.-Vereinigung, Bd. VIII., 1900. For applications of the theory to analysis, see Jordan's Cours, quoted above, and Borel, Le(ons sur la Thiorie des Fonctions, Paris, 1898. For Veronese's conception of the continuum, see his book Fondamenti di geometria, Padua, 1891 (German translation, Leipzig, 1894).

PART II

ALTERNATIVE THEORIES OF REAL NUMBERS

DOUGLAS S. BRIDGES

A CONSTRUCTIVE LOOK AT THE REAL NUMBER LINE

1.

INTRODUCTION

Our aim in writing this paper is to present some of the distinctive features of the real number line R• as it appears to the constructive mathematician. Throughout this presentation we shall pay particular attention to constructive notions and proofs that differ from their classical counterparts, or whose classical analogues are insubstantial (in the case of notions) or trivial (in the case of proofs). For example, we explain why one classical definition of 'closed subset of R' is inappropriate in the constructive setting (6.2); and we devote a considerable amount of space to the property of locatedness, which plays no role whatsoever in traditional analysis (Section 12). We shall only consider real numbers constructed from Cauchy sequences of rational numbers; we shall not be concerned with Dedekind reals (Staples, 1971), extensions of the constructive continuum (Troelstra, 1980; Troelstra, 1982), or embeddings of the constructive continuum in the classical one (Lifschitz, 1982). Before going any further, we explain what we mean by constructive mathematics and discuss briefly the three varieties of constructive mathematics that are currently considered to be of most significance for philosophy, mathematical logic, and computer science. In doing so, we shall make no attempt to describe in detail, let alone justify, any constructivist philosophy of mathematics.' Nor, apart from sketching the outlines of intuitionistic logic, shall we become involved in those aspects of constructivism that can be regarded as part of mathematical logic, rather than mathematics proper.2 What, then, distinguishes constructive mathematics from its traditional, or classical, counterpart? The distinction rests primarily upon the constructive mathematician's strict interpretation of existence: whereas in classical mathematics it is common to prove the 'existence' of an object x with a property P by deducing a contradiction from the assumption that no such x exists, a constructive proof of the existence of such an x must embody both an algorithm for the construction of x (at least to 29 P. Ehrlich (ed.), Real Numbers, Generalizationsof the Reals, and Theories of Continua, 29-92. © 1994 Kluwer Academic Publishers. Printed in the Netherlands.

30

DOUGLAS S. BRIDGES

any preassigned degree of accuracy) and an algorithm which verifies that x has the property P. Thus, in constructive mathematics, existence means computability. Following Bishop, we shall take the notion of algorithm as primitive; but there is no reason why the reader, if he so wishes, should not interpret an algorithm as, for example, a syntactically correct program written in a programming language of his choice. Whatever interpretation one makes of the word 'algorithm', however, it should only admit computations that can be performed by a finite number of human beings or computers in a finite length of time. Note that our algorithmic interpretation of existence makes no demands about the complexity or efficiency of the algorithms we use. It would be fascinating, and certainly extremely challenging, to develop constructive mathematics with careful attention to questions of complexity, but such a development is probably several stages beyond anything that constructive mathematicians have so far accomplished; at present, constructive mathematics addresses questions of computability in principle, rather than computability in practice (see, however Ko (1991)). If algorithmic method is characteristic of constructive mathematics, so also is numerical content: according to Bishop, The primary concern of mathematics is number, and this means the positive integers. We feel about number the way Kant felt about space. The positive integers and their arithmetic are presupposed by the very nature of our intelligence and, we are tempted to believe, by the very nature of intelligence in general. The development of the theory of the positive integers from the primitive concept of the unit, the concept of adjoining a unit, and the process of mathematical induction carries complete conviction .... Building on the positive integers, weaving a web of ever more sets and more functions, we get the basic structures of mathematics: the rational number system, the real number system, the euclidean spaces, the complex number system, the algebraic number fields, Hilbert space, the classical groups, and so forth. Within the framework of these structures

most mathematics is done. Everything attaches itself to number, and every mathematical statement ultimately expresses the fact that if we perform certain computations within the set of positive integers, we shall obtain certain results (Bishop, 1967, pp. 2-3)

We have already alluded to the three main varieties of modern constructive mathematics. The first of these, which provides the framework for the remainder of this paper, is Bishop's constructive mathematics, hereafter referred to as BISH. In this variety the notion of algorithm is primitive, and there is no commitment to any formal system or to special

A CONSTRUCTIVE LOOK AT THE REAL NUMBER LINE

31

principles (such as the Church-Markov-Turing thesis). In consequence, every proof of a proposition P within BISH is both a proof of P in classical mathematics and a proof of the appropriate interpretations of P in the other two varieties of constructive mathematics that we are about to describe. Each of these two varieties can be regarded formally as BISH together with certain additional principles. In the first variety, the recursive constructive mathematics (RUSS) of the Russian school of Markov, the main principle adjoined to BISH is a form of the Church-Markov-Turing thesis that all sequences of natural numbers are recursive. In the other, Brouwer's intuitionistic mathematics (INT), there are adjoined to BISH two principles which ensure strong continuity properties of functions on intervals of the real number line. As many readers will be aware, to view RUSS and INT formally as extensions of BISH is to ignore the foundational issues which gave birth to all three varieties. In RUSS every mathematical object is, at heart, a natural number, and constructions are set in a specific formal system within which, for example, functions are Gbdel numbers of the algorithms that compute them. On the other hand, INT is firmly rooted in Brouwer's philosophy of intuitionism, which involves, amongst other matters, an analysis of the notion of 'free choice sequence'; this, in turn, produces the two principles which lead to the strong continuity properties referred to above. In the rest of this paper we shall deal only briefly with the continuum in RUSS, and not at all with the intuitionistic continuum, which is discussed in detail in such references as Brouwer (1967), Dummett (1977), Kleene and Vesley (1965), and Troelstra and van Dalen (1988). For a discussion of the not entirely straightforward relation of RUSS and INT with classical mathematics, see Chapter 6 of Bridges and Richman (1987). In all varieties of constructive mathematics the algorithmic interpretation of existence forces one to reconsider the meaning of each logical connective and quantifier.3 For example, and in contrast to the classical view that every mathematical statement is either true or false (even if we cannot say which alternative holds), the constructive mathematician does not consider a proposition P to be true or false unless he can produce either a proof of P or a proof that P is impossible. Thus the law of excluded middle, which asserts that for any statement P either P or its negation is true, cannot be accepted within constructive mathematics.

32

DOUGLAS S. BRIDGES

The rejection of the law of excluded middle can be clarified by considering a special case, applicable to binary sequences. A binary sequence (aJ)is an algorithm which, applied to any positive integer n, produces an output an equal to either 0 or 1; it is implicit in this definition that for each n we can decide whether a. = 0 or a, = 1. Bishop has called the following classically trivial statement about binary sequences the limited principle of omniscience: LPO

If (aJ)is a binary sequence, then either there exists n such that an = 1, or else a. = O for all n.

A constructive proof of LPO could be converted into an algorithm which, applied to any binary sequence (an), would compute an integer N with the following properties: N> 0 N=

if and only if there exists n such that an which case aN = 1; I if and only ifan = 0 for all n.

=

1, in

Suppose we have such an algorithm, and apply it to the binary sequence (aJ)defined as follows: a, = 0 = I

if 2k is a sum of two primes for each positive integer k
Note that as, by a finite amount of testing, we can determine for any positive integer n whether or not 2k is a sum of two primes for each positive integer k < n, the sequence (a.) is constructively well defined. If our algorithm for proving LPO outputs -1, and therefore demonstrates that a, = 0 for all n, we conclude that the Goldbach conjecture every even positive integer is a sum of two primes is true; if our algorithm outputs a value N for which aN = 1, then, by testing 2k for each positive integer k < N, we can find a counterexample to the Goldbach conjecture. So a constructive proof of LPO would provide an algorithm for deciding the Goldbach conjecture. Similar arguments show that a constructive proof of LPO would lead to decision procedures for many unsolved problems of mathematics, including the Riemann hypothesis and the Fermat conjecture. The availability of such a powerful algorithm as one that proves LPO therefore

A CONSTRUCTIVE LOOK AT THE REAL NUMBER LINE

33

seems highly unlikely; moreover, LPO can be proved false in RUSS and INT.4 We are therefore led to reject LPO from the constructive canon. Another classically trivial principle which is rejected for similar reasons is the lesser limited principle of omniscience: LLPO

If (an) is a binary sequence containing at most one term equal to 1, then either an = 0 for all even n, or else a. = 0 for all odd n.

Any classical proposition that constructively entails LPO, LLPO, or some other non-algorithmic principle (like the principle of finite possibility, introduced in Section 2 below) is considered to be essentially nonconstructive, and is outcast from constructive mathematics. As we shall see later, among such propositions are the law of trichotomy for each real number x, x > 0 or x = 0 or x < 0, which is equivalent to LPO, and the statement for each real number x, either x > 0 or x < 0, which is equivalent to LLPO. There is another classically trivial proposition whose role in constructive mathematics is more controversial than that of LPO or LLPO. This proposition is accepted, with some qualms, by practitioners of RUSS, but not by proponents of BISH and INT. The proposition in question is Markov's principle: MP

If for a binary sequence (an), it is impossible that all the terms equal 0, then there exists (that is, we can find) a value m such that am = 1.

One attempt to justify MP says that if the hypotheses of MP hold, then, by examining the terms a,, a2.... of the binary sequence one by one, we are guaranteed to find a value n such that a. = 1. What makes most constructive mathematicians suspicious of MP is that it does not provide us, in advance of such an examination of terms, with a bound to the number of terms that we must look at before we can be sure that we have found one equal to 1. For this reason, most constructivists either reject outright, or refrain from using, any propositions that are constructively equivalent to MP.5

34

DOUGLAS S. BRIDGES

LPO and LLPO illustrate not only the constructive interpretation of the existential quantifier 3 (there exists), but also that of the connective V (or): to justify the assertion P1 V P 2 for propositions P1 and P2 , we must have either a proof of P, or a proof of P,. In other words, P1 V P 2 is equivalent to 3i Pi. The constructive interpretation of A (and) and V (for each) pose no problem: to prove P, A P 2 we must produce both a proof of P1 and a proof of P 2; to prove VxP(x) we must produce an algorithm which, applied to any object x in the universe of discourse, proves that P(x) holds. 6 The commonest constructive interpretation of the connective (implies) can be paraphrased as follows: P = Q means that Q holds under the assumption that P holds, or that we can derive Q from the hypothesis P. As Bishop says, The validity of the computational facts implicit in the statement P must ensure the validity of the computational facts implicit in the statement Q. (Bishop and Bridges, 1985, p. 10)

However, the reader should be aware that Bishop and others have expressed dissatisfaction with such an interpretation of * (Bishop, 1970); it is therefore most unfortunate that Bishop left only rudimentary notes as a distillation of his many years of reflection on this topic. The statement -,P (not P) is interpreted as P = Q, where Q is a contradiction (typically, 0 = 1). The standard classical interpretation of the connective = is 'material implication', in which P => Q is, by definition, -,P V Q. Certainly, we can derive Q constructively from the two hypotheses -,P V Q and P, so that (-,P V Q) := (P • Q); but as P * P holds trivially and we cannot prove the law of excluded middle, -,P V P, constructive implication and material implication are not equivalent constructively. Another classical principle rejected by constructive mathematicians is -,P = P: for, taking P to be of the form 3xQ(x), we see that if -,--,P holds, then it is absurd to deny that there exists x with the property Q(x); but such an absurdity does not provide us with a algorithm for constructing x such that Q(x) holds.' However, Brouwer has shown that --,P: for if P holds, then so does --,--,P; and, in we can prove 1 P constructive as in classical mathematics, if A => B, then -,B = -A. Having briefly disposed of the background to, and the logic underlying, the work in this paper, we turn in the next section to the

A CONSTRUCTIVE LOOK AT THE REAL NUMBER LINE

35

mathematical building blocks with which the real line and other objects of constructive mathematics are created. 2.

BASIC CONSTRUCTIONS

The ability to construct collections of objects that have been, or could have been, already constructed is an essential feature of modern mathematics, as is the ability to place objects from one such collection in partial correspondence with objects from another. The construction of set S consists of two parts: (i) an explanation of how we construct elements (members) of S using objects that have been, or could have been, constructed prior to S; and (ii) an explanation of what it means for two elements of S to be equal. The equality, =, on a set is an essential part of its description, and must satisfy the defining properties of an equivalence relation: reflexivity symmetry transitivity

x = x; if x = y, then y = x; if x = y and y = z, then x = z.

We write x E S to mean that x is an element of S. When the elements of a set can be written in a finite or infinite list, we may denote the set by writing within braces a list, or an indication of a list, of its elements: for example, {0, 1, 2, . . .) represents the set whose members are the natural numbers 0, 1, 2 .... The equality relation on the set N' of positive integers, and that on the set N of natural numbers (nonnegative integers), is the relation of identity: two natural numbers are equal if and only if they are one and the same. A property P which is applicable to the elements of a set A determines a subset S of A denoted by {x E A : P(x)}: if x is an element of A, then x E S if and only if P(x); and the equality on S is the restriction to S of the equality on A. Note that we are only concerned with properties P(x) that are extensional, in the sense that for all x, x' in A with x = x', P(x) if and only if P(x'); we are not concerned with intensional properties - those that depend on the manner in which objects are presented to us. If S is a subset of A, and x ( A, we write x 0 S to mean that x E S is impossible; and we denote by A\S the set {x - A : x v S}.

36

DOUGLAS S. BRIDGES

Another basic construction of mathematics is that of an ordered pair (a, b), where a belongs to a set A and b to a set B. Taken with the equality (x, y) = (x', y') if and only if x = x' (in A) and y = y' (in B), the set of all ordered pairs (a, b) with a E A and b e B is called the Cartesianproduct of A and B, and is written A x B. A subset of A x A, or, equivalently, a property applicable to elements of A x A, is called a binary relation on A. A set is nonempty, or nonvoid, if we can construct an element of it. An empty set is a set that cannot be nonempty; we denote the empty subset of a set S by 0, or, when no confusion is likely, simply by 0. To show that a set S is nonempty, it is not enough to prove that a contradiction arises from the assumption that S is empty: we must show how to construct, at least in principle and with any preassigned accuracy, an element of S. If S, and S 2 are subsets of a set A such that x e S2 whenever x ( S, we say that S, is contained, or included, in S 2, and that S2 contains, or includes, $I; we then write either S, C S2 or S2 D S 1. Two subsets of A are equal if each is contained in the other. A function from a set A to a set B is an algorithm f which produces an element f(x) of B when applied to an element x of A, and which has the property that f(x) = f(x') whenever x = x'; thus functions, like properties, are extensional. A function is also called a mapping, or a map. The notation f : A -ý B indicates that f is a function from A to B; the set A is called the domain of f, and is written dom(f). For each subset S of A we let f(S) be the subset

{y e B : y

=

f(x) for some x E S} = {f(x) : x E S)

of B; the set f(A) is called the range of f. If the range of f equals B, we say that f maps A onto B; if x = x' whenever f(x) = f(x'), then f is one-one. A partialfunction f from a set A to a set B, or (by abuse of language) a partialfunction f : A -* B, is a mapping whose domain is a subset of A and whose range is a subset of B. We say that f(x) is defined if x e dom(f), and undefined if x t dom(f). A partial functionf: A -* B whose domain is the entire set A is said to be total, or, oxymoronically, a total partialfunction f : A -- B. The notion of partial function is of great importance in computability theory; we shall use it in our discussion of the Church-Markov-Turing thesis and its consequences.

A CONSTRUCTIVE LOOK AT THE REAL NUMBER LINE

The composite of two partial functions f : A the partial function g of : A -4 C defined by 8 g of(x)

-ý

37

B and g : B -- C is

g(f(x))

wherever the right side exists. A subset S of a set A is detachable from A if for each x in A either x E S or x 0 S. A detachable subset S of A may be identified with the function f from A to {0, 1) such that f(x) = 1 if and only if x E S. For any set A, both A and OA are detachable from A. An (infinite) sequence x is an algorithm which associates an object x, - the nth term of the sequence - with each positive integer n; we usually denote a sequence x by (x,)-=, or simply (xJ). If each term x, belongs to a fixed set A, then the sequence (x,) is a function from N' to A. A subsequence of (xn) consists of (x,) and a sequence (n,)., of positive integers such that n, < n2 < . . ; we identify this subsequence with the sequence whose kth term is x,k. Sometimes we consider a sequence (x,,)=,, defined on N, rather than N'+; when no confusion is likely, we shall denote such a sequence also by (x,,). A finite sequence of length n, where n E N', is a mapping x with domain { 1, 2 .... n}; this mapping can be identified with the ordered n-tuple (x ..... x,), where xi - x(i). Let (A,) be an infinite sequence of subsets of a set S. We define the union of that sequence to be U-,An =-= {x E S : x E A. for some n}, and the intersection to be

n-=,A,-

(x E S : x E An for all ni.

Note that the union and intersection of (A.) are subsets of S; and that the union A, U A2 U ... U A., and intersection A, N A2 n ... N An, of a finite sequence (A ..... A,) of subsets of S are defined in the obvious analogous way. If P(n) is a property of positive integers n, we also define U{AI : P(n)}I A{A, :P(n)}

{x E : 3n(P(n) A x E AJ)} {x c S: Vn(P(n) - x E A,)}

Unions and intersections have many, but not all, of the properties familiar from classical mathematics. For further details, see Bishop and Bridges (1985, Ch. 3, Section 2).

38

DOUGLAS S. BRIDGES

Two subsets A, B of R are disjoint (from each other) if A f B = 0. A set A is finitely enumerable, or subfinite, if there exist a nonnegative integer n and a map f from {k e N' : k < nI onto A; if the map f is also one-one, we say that A is finite. The reader may show that a finitely enumerable subset of N is both finite and detachable; and that a set A is finitely enumerable if for some positive integer n there exists a mapping from a detachable subset of {k e N' : k < n} onto A. (2.1) An empty set is finite (and therefore finitely enumerable). Proof. If A is an empty set, then the partial function f : N' A U {J1 } defined by f(n) a 1 if and only if n < 0 maps the empty set {k e•N' : k • 01 onto A. Hence A is finite.

a

(2.2) If D C N is detachable and F C N is finite, then D n F is finite. Proof. Construct a nonnegative integer n and a one-one map f of {k E N' : k < n} onto F. Since D is detachable, the set A

{k E N' : k • n and f(k)

E

D}

is detachable. Defining g(k) =-f(k) for all k in A, we obtain a mapping of A onto D n F. Hence D n F is finitely enumerable and therefore finite. M A set A is countable if there is a function from a detachable subset of N onto A. As in classical mathematics, the Cartesian product N x N is countable: in fact, there exists a one-one map of N onto N x N. (2.3) An empty set is countable; a nonempty set is countable if and only if it is the range of a function with domain N. Proof. If A is an empty set, then the function f: ON -- A U {I1} defined by f(n) -= 1 if and only if n E O maps ON onto OA" Since ON is detachable from N, it follows that OA is countable. Now consider a nonempty countable set A. Constructing a E A and

A CONSTRUCTIVE LOOK AT THE REAL NUMBER LINE

39

a function f from a detachable subset D of N onto A, define a function g : N - A as follows: g(n) =f(n) =a

if n E D, if n D.

Then g maps N onto A. On the other hand, as N is detachable from N it is immediate that the range of a function from N is countable. * In view of (2.3), the elements of a nonvoid countable set A can be listed in an infinite sequence a 1, a 2.... ; such a list is called an enumeration of A. A sequence (An) of subsets of S is increasing if An C A,, 1 for each n. The following characterisation of countable subsets of N is useful in computability theory and recursive constructive analysis. (2.4) The following are equivalent conditions on a subset S of N. (i) S is countable. (ii) S is the union of an increasing sequence of finite subsets. (iii) S is the union of a sequence of finite subsets. Proof. If S is countable, andf is a mapping from a detachable subset D of N onto S, then for each nonnegative integer n define Dn=_D

n {0, 1....

n}.

By (2.2), Dn is finite; also, clearly, D, C Dn,•, and S, -f(D,) is finite, Sn C Sn.l, and S = Un.iS_. Thus Clearly, (ii) implies (iii). To complete the proof, the union of sequence (Sn)-=, of finite subsets. Define f: N x N -- N as follows: f(m, n) = m = undefined

D = U-=,Dn. So (i) implies (ii). suppose that S is a partial function

if m E S, otherwise.

Then the range of f equals S. Also, since each Sn is finite and therefore detachable from N, the domain of f is detachable from N x N. If g is a one-one map of N onto N x N, then D =- {n E N : g(n) c dom(f)}

is detachable from N, and the function fog maps D onto S.

N

40

DOUGLAS S. BRIDGES

Perhaps surprisingly, we cannot expect to prove that a nonvoid subset of N is countable. Before justifying this claim, we state the principle of finite possibility: PFP

To each binary sequence (an) there correspondsa binary sequence (bn) such that Vn(an

=

0) if and only if

3n(b.

=

1).

The interested reader may show that PFP and Markov's principle are together equivalent to LPO. Since LPO is false, and Markov's principle is accepted, in RUSS, we do not expect to produce a constructive proof of PFP, (2.5) The statement every nonvoid subset N is countable entails PFP. Proof. Let (a.) be a binary sequence with a, = 0, and let J- {1} U {n E N : Vk(n = ak

+

I)}.

Then J is a subset of N containing 0, and 1 E J if and only if ak = 0 for all k. It follows immediately that if J is countable and b1, b2.... is an enumeration of J, then an some n. 3.

=

0 for all n if and only if bn

=

1 for U

THE AXIOM OF CHOICE

There seems to be a widespread belief that a constructive mathematician can be identified by his rejection of the axiom of choice: AC

If S is a subset of A x B, and for each x in A there exists y in B such that (x, y) E S, then there is a function f from A to B called a choice function for S - such that (x, f(x)) E S for each x in A.

This belief is, at best, deficient: as is evident from the discussion in Section 1 and the quotation from Bishop on page 30 a constructive mathematician is identified by the attention he pays to the meaning

A CONSTRUCTIVE LOOK AT THE REAL NUMBER LINE

41

of mathematical expressions, especially the phrase 'there exists'. Nevertheless, the axiom of choice, in the stated form, is unacceptable within BISH: (3.1) The axiom of choice entails the law of excluded middle. Proof. Let P be any constructively meaningful statement. Let A {0, 1}, with equality defined by 0 = I if and only if P; let B = {O, 1), with the standard equality; and let S be the subset {(0, 0), (1, 1)} of A x B. If f: A -- B is a choice function for S, then we have two possibilities: (i) either f(O) = 1 or f(1) = 0; (ii) f(O) = 0 and f(1) = 1. In case (i), if, for example, f(O) = 1, then (0, 1) e S. So either (0, 1) = (0, 0), which forces 0 and 1 to be equal elements of N and is therefore impossible; or, as must be the case, (0, 1) = (1, 1). Thus 0 and 1 are equal elements of A, and so P holds. On the other hand, if P holds, then as 0 and I are equal elements of A, and as f is a function, we have 0 f(0) =f(1), which rules out case (ii). Hence in case (ii) P is false. The above proof was first presented in Goodman and Myhill (1978); however, it is reasonable to assume that Bishop was aware of (3.1) during the writing of his 1967 book (see Bishop, 1967, Ch. 2, Problem 2). Although AC is not part of BISH, a restricted form of choice - the principle of dependent choice - is generally accepted and widely used by constructive mathematicians: DC

If a E A and S C A x A, and for each x in A there exists y in A such that (x, y) E S, then there exists a sequence of elements a,, a2, . . . of A such that a, = a and (an, an,,) E S for each positive integer n.

A consequence of DC is the principle of countable choice, which is the case A = N of AC.

42

DOUGLAS S. BRIDGES 4.

THE REAL NUMBERS

Passing 6ver the standard construction of the set Z of integers, we define a rational number to be an ordered pair (m, n), usually written m/n, of integers such that n # 0; two rational numbers m/n and m'/n' are equal if mn' and m'n are equal integers. The familiar algebraic operations and order relations on the set Q of rational numbers behave constructively as they do in classical mathematics. We identify the integer n with the rational number n1l. We could carry over into the constructive context the standard classical definition of a real number in terms of general Cauchy sequences of rational numbers; but such a procedure would involve our coupling each rational Cauchy sequence (xn) with a sequence (nk)7=1 of positive integers giving the rate of convergence of (x,): for example, we might require the terms n, to satisfy Ix. - x.j < 1/k for all m, n > n,. It is, however more convenient to single out a special type of rational Cauchy sequence in our definition of constructive real numbers. It is also convenient to call a sequence of this special type, rather than an equivalence class of such sequences, a real number: for, to specify an equivalence class constructively, we need to provide a representative member of that class; so we might as well concentrate on the Cauchy sequence that does that specification, rather than on the entire class so specified. Here is the formal definition. A real number is a sequence x - (xn)n=l of rational numbers that is regular, in the sense that IX, - xn1 < 1/rm + l/n for all positive integers m and n; the term x, is called the nth rational approximation to the real number x. We identify a rational number r with the real number (r, r, r, ... ). To specify completely the set R of real numbers, we must equip it with an appropriate notion of equality: two real numbers x a (x,) and y a (y.) are equal if Ixn - Y.J 5 2/n for each positive integer n. Note that this notion of equality is an equivalence relation: it is clearly reflexive and symmetric, and its transitivity is a simple consequence of the following lemma. (4.1) Two real numbers x (xn) and y (yj)are equal if and only if for each positive integer k there exists a positive integer Nk such that Ix,, - y[j < I/k whenever n > Nk. Proof If x = y, then for each k we need only take Nk = 2k. Conversely, suppose that for each k there exists N, with the stated property, and

A CONSTRUCTIVE LOOK AT THE REAL NUMBER LINE

43

consider a positive integer n. For any positive integers m, k with m > max{k, Nk} we have

IXn - YnI < IXn - XmI + IXm - YmI + 1Yi- YnI < (1/n + I/m) + I/k + (1/n + 1/m) < 2/n + 3/k. Since this holds for all positive integers k, we see that Ix. - yj, < 2/n; whence, as n is arbitrary, x = y. 0 Note that the algorithm which assigns to each real number its nth rational approximation is not a function relative to the equality relations on R and Q: for example, the real numbers x = (1, 1/2, 1/3 .... ) and y = (0, 0, 0, . . .) are equal, but their nth rational approximations are not. To introduce the algebraic operations on R we need an appropriate bound for all the terms of a regular sequence x -=(x,,) of rational numbers. The canonical bound K. of x is the least positive integer greater than lxli + 2; it is easy to show that Ix&I < g4 for all n. The arithmetic operations on real numbers x - (x,) and y (yn) are defined in terms of the rational approximations to those numbers, as follows: (x + y).

(xy)., max{x, y),} min{x, y}),

X2.n± Y2n

X2kJY2kn, where k max{xn, yJ} min{x., yj}

max{Kx, KY}

IxI - IxnI,

where, for example (x + y)A denotes the nth rational approximation to the real number x + y, and max{x.,, y•j is the maximum, computed in the usual way, of the rational numbers x. and yn. Of course, we must verify that the above definitions do provide us with real numbers; we illustrate this verification with the case of the product xy. Writing z, =-X2kMY2kn - so that xy = (z.) - for all positive integers m and n we have

IZm - zn1 = IX2km(Y2km - Y2kn) + Y2kn(X2km - X2kn)I - kly 2kn - Y2knI + klX 2 km - X2kn1 - k(1/2km + 1/2kn) + k(1/2km + 1/2kn) = 1/m + 1/n. Thus xy is a regular sequence of rational numbers - that is, a real number.

44

DOUGLAS S. BRIDGES

The arithmetic operations defined above obey most of the rules familiar from classical mathematics. Note, however, that some classical rules of arithmetic do not pass over to the constructive setting: (4.2) The statement Vx, y E R (xy = 0 • x = 0 or y = 0) entails LLPO. Proof. Assume that the statement in question holds, and consider an arbitrary binary sequence (aj)with at most one term equal to 1. Define xn= 1/k = 0

ifk n, kisodd, andak= 1, otherwise

yý = Ilk =0

if k < n, k is even, and ak = 1, otherwise.

and

Then x a (x,) and y = (y,) are regular sequences of rational numbers; moreover, as xyn = 0 for all n, xy = 0. If x = 0, then a. = 0 for all odd n; and if y = 0, then a, = 0 for all even n. Since (an) was any binary sequence with at most one term equal to 0, we conclude that a proof of the statement Vx, y E R (xy = 0 => x = 0 or y = 0) can be converted, U constructively, into a proof of LLPO. A real number x - (xv) is positive if there exists n such that xn > 1/n. If y is also a real number, then we define x > y to mean that x - y is positive; thus x > 0 if and only if x is positive. On the other hand, we say that x is negative nonnegative

if -x is positive, if xn > -1/n for all n;

and we write x > y to denote that x - y is nonnegative. We define x < y and x < y to have the obvious meanings relative to the relations > and >. (4.3) A real number x =_(x,) is positive if and only if there exists a positive integer N such that xm > 1/N for all m > N. On the other hand, x is nonnegative if and only if for each positive integer k there exists a positive integer Nk such that x_ > -Ilk for all m Ž_Nk. Proof. If x is positive, then x. > 1/n for some n. Choosing the positive integer N so that 2/N < x_ - l/n, whenever m _ŽN we have

A CONSTRUCTIVE LOOK AT THE REAL NUMBER LINE

45

x. Ž > X. - Ixn - x.1 > x - l/m - 1/n 2_x, - 1/n - 1/N > 1/N. So the required property holds. If, conversely, that property holds, then x,+, > 1I(N + 1), so x > 0. The proof of the second part of the proposition is left to the reader. U Another useful result is the following. (4.4) Two real numbers x and y are equal if and only if Ix - yI < , for each positive real number P. Proof. Suppose Ix - yI < E for each e > 0, and let k be any positive integer. Since Ix - YI < I/k, it follows from (4.3) and the relevant definitions that there exists N such that

1/k - Ix,,

-

Y4nI Ž 1/N

for all n > N. For all n > 5N/2 we then have Ix, - YnJ < Ix, - x4 .1 + Ix4. - Y4nI + IY4n - YnI < (1/n + 1/4n) + (1/k - 1/N) + (1/4n + l/n) = 5/2n - 1/N + 1/k 1/k. Hence x = y, by (4.1). The proof of the converse is left to the reader. U The classical law of trichotomy Vx, yE R (x>yorx=yorx 0 => x > 0 or x = 0) is equivalent to LPO. Proof. Suppose the statement in questions holds, and consider any binary sequence (an). Define xn = 1/k = 0

ifk n, aj= 0forallj
It is a simple exercise to show that x - (xn) is a nonnegative real number.

46

DOUGLAS S. BRIDGES

If x > 0, then there exists N such that XN > 1/N; by inspection of the terms a, . . . . . aN we can then find k < N such that ak = 1. If x = 0, then a, = 0 for all n. As (aj)is an arbitrary binary sequence, we have shown that a proof of the statement in question can be converted, constructively, into a proof of LPO. Conversely, suppose that LPO obtains, and consider any real number x = (x,). Since each x. is rational, we can define a binary sequence (a.) such that an = 0 '=- X < 1/n and a. = I

'=- xn >

1/n.

By LPO, either a. = 0 for all n or else there exists N such that a, = 1. In the former case, for each positive integer n we have -x, > -1/n; so -x > 0 and therefore x < 0. In the case where a, = I for some N, XN >

I/N and therefore x > 0.

U

The reader may prove (4.6) Each of the following statements is equivalent to LLPO: (i) Vx E R (x >ŽO or x < 0); (ii) for all real numbers x, y either max{x, y} = x and min{x, y} = y, or max{x, y} = y and min{x, y} = x. 0 The second part of (4.6) has an interesting consequence in the theory of polynomial factorisation: since, as is easily verified, (x - a)(x - b) = (x - max{a, b})(x - min{a, b}) for all a, b

E

R, the statement

if a, b, c are solutions of a quadratic equation over R, then either c = a or c = b entails LLPO. In other words, we cannot expect to prove the unique factorisation of polynomials over R! Note, however, that we are not saying that a quadratic equation over R can have three distinct solutions: it is straightforward to prove constructively, as classically, that the existence of three distinct solutions of a quadratic equation is contradictory. What we are saying is that we do not expect to produce an algorithm which, applied to three real numbers that satisfy a given

A CONSTRUCTIVE LOOK AT THE REAL NUMBER LINE

47

quadratic equation, will enable us to find two of those numbers that are equal. Of the several important results that we shall derive about the orderings on R, the next is, in view of its use of contradiction, perhaps the most unexpected. (4.7) If x and y are real numbers such that the assumption x > y leads to a contradiction, then x < y. Proof. It will suffice to consider the case where y = 0. Let x - (x), and consider any positive integer n. Since the law of trichotomy holds for rational numbers, either xn > 1/n or x_ < l/n. The former alternative is ruled out, since it entails x > 0. Thus we have x_ < I/n, and therefore -xn > -1/n, for all n. So -x > 0 and therefore x < 0. 0 The constructive status of the contrapositive of the last result is not so clear: (4.8) The statement

Vx c= R (-,(x <_0) => x > 0) is equivalent to Markov's principle. Proof. Suppose the statement in question holds, and consider any binary sequence (an) such that -Vn(an = 0). Define a real number x (x,) by setting x. = 1/k =0

ifk!n, aj= 0 for allj< k, and otherwise.

ak=

1,

Suppose that x < 0; then -x_ > -1/n for all n. If also there exists m such that am = 1, and if k is the least such m, then -x,+, = -1/k < -1/(k + 1),

a contradiction. Thus Vm(am = 0), again a contradiction. So -I(x < 0). Hence, by our initial assumption, x > 0. Choosing N so that xN > I/N, we see that ak = 1 for some (determinate) k < N. As (a.) was any binary sequence satisfying -Vn (an = 0), we conclude that Markov's principle holds. Now consider any real number x - (xn) such that -,(x < 0). Since each x& is rational, we can define a binary sequence (an) such that

48

DOUGLAS S. BRIDGES

a= 0 4=- x,< 1/n, o a.= 1 * x. > 1/n.

If a, = 0 for all n, then -x_ > -1/n for each positive integer n; so -x > 0 and therefore x < 0. This contradiction ensures that -,Vn(a. = 0). If Markov's principle holds, we can now find n such that a. = 1; whence x, > 1/n and therefore x > 0. Thus Markov's principle implies that Vx

R(-,(x<0) =x>0).

U

As we have already pointed out, within BISH there is no known proof of, or counterexample to, Markov's principle, so we cannot use the principle Vx E R (-n(x _ 0) = x > 0). However, Markov's principle, and therefore the statement under discussion, is accepted by practitioners of RUSS. Two real numbers x, y are unequal, or apart, if Ix - YI > 0; in which case we write x # y. A real number x is nonzero if x # 0. The reader may show that the statement Vx E R (-(x = 0) • x # 0) is equivalent to Markov's principle. (4.9) The following properties hold for real numbers x and y: (i) if x > 0 and y > 0, then x + y > 0; (ii) max{x,y} >0if and only ifx>0ory>0; (iii) ifx+y >O, then eitherx> 0 ory >O; (iv) if xy # 0, then either x # 0 or y * 0; (v) if x > y, then for all real numbers z either x > z or z > y. Property (v) is particularly important as a partial substitute for the classical law of trichotomy. We shall prove only (iii), (iv), and (v); this will require some preliminary observations. First, we note that if x - (x,) is a real number, then Ix - x.1 < 1/n for all n: for, by the relevant definitions, the mth rational approximation to 1/n - Ix - x.1 is 1/n

-

IX 4 m -

Xn1

Ž 1/n - (1/4m + l/n) = -114m > -1/m;

so 1/n - Ix - xjl > 0 and therefore Ix - xl < 1/n. Next, we prove that Q is orderdense in R: that is, for all real numbers x, y with x < y there exists a rational number r such that x < r < y. Indeed, taking x =-(x,) and y (y.), we have 0
x = (Y2ý

-

X2n.=l,

A CONSTRUCTIVE LOOK AT THE REAL NUMBER LINE

so there exists N such that yw

x2N > 1/N. Writing r

-

= 2(X2N + Y2N),

49

we

have r - x > r - xv-

Ix2

- xI

1/2N > 0,

> Ž(Y2N - X2N) -

and similarly y - r > 0; hence x < r < y. We can now prove statements (iii) and (v) of (4.9) above. If x a (x,) and y = (y.) are real numbers with x + y > 0, then there exists a rational number a such that 0 < ax < x + y. Choose a positive integer n > 4/ac, and let r - x,, s a y.. Then r, s are rational, Ix - rl < W/4, and ly- sI < ca/4. So r + s > (x + y) - (Ix - rl + ly > a - (cx/4 + (/4) = a/2.

-

sI)

Since r and s are rational numbers, either r > a/4 or s > ca/4. In the first case, x > r - Ix - rl > 0; in the second, y > 0. To prove property (v), consider any real numbers x, y with x < y. For all z E R we have (x

-

z) + (z

-

y) =x

-

y> 0;

so, by (iii), either x - z > 0 or z - y > 0. For a nonzero real number x, the reciprocal, lix (or x-'), is defined as follows. Choose a positive integer N such that IxI 2! 1/N for all n _>N, and define l/x by (l/x), = 1/XN3 = l1/XnN2

if n < N, if n > N.

We omit the details of the proof that l1x is the unique real number t such that xt = 1, and that the function taking a nonzero real number x to lix maps the set of nonzero real numbers onto itself and has the expected algebraic properties [Bishop and Bridges, 1985, Ch. 2, (2.13)]. Instead, we end this section with the deferred proof of part (iv) of (4.9) above. Let x, y be real numbers with xy # 0. Replacing x by IxI/IxyI and y by lyl/Ixyl, we can assume that x > 0, y > 0, and xy = 1. Then either x > 0 or x < 1/2. In the latter case, if y < 1 we have the contradiction xy < 1/2; so y > 0. This completes the proof of (4.9, iv).

50

DOUGLAS S. BRIDGES 5.

COMPLETENESS

A sequence (x,) of real numbers converges to a real number x_, called the limit of (x,), if for each positive integer k there exists a positive integer Nk such that Ix, - x- < 1/k whenever n > Nk; in that case, we write

or

xn -4

x- as n -4

oo.

Note that if (x,) converges to both x- and x'-, then x. = x'-. For the usual results about limits of sums, products, and so on, the reader should consult Bishop and Bridges (1985, Ch. 2, (3.4)). A convergent sequence (x,) is a Cauchy sequence: that is, for each positive integer k there exists a positive integer Nk such that IXm - x.1 < 1/k for all m, n > Nk. A subset S of R is complete if each Cauchy sequence in S converges to a limit that belongs to S. There is a misconception among many classical mathematicians that the constructive number line must be incomplete. This misconception may be due, in large part, to an impression that applications of completeness enable us to prove convergence without actually showing how to find the limit. It may also be due to an incomplete understanding of the phenomenon first discovered by Specker: there exists a strictly increasing sequence of rational numbers between 0 and 1 that does not converge recursively; see (11.1) below. Classically, such a sequence is a Cauchy sequence and so converges; of course, its classical limit is not a recursive real number. But in the strictly recursive context that sequence is not a Cauchy sequence - it cannot be, as it does not converge; so it does not provide a counterexample to the completeness of the recursive real line. In fact, the recursive real line is complete, because the proof within BISH of the following result carries over mutatis mutandis into the recursive context. (5.1) The set R of real numbers is complete. Proof. Consider any Cauchy sequence (x,) in R. For each positive integer k choose Nk in N+ such that IXm - Xn1 < 1/k whenever m, n > Nk, and write v(k) - max{3k,

N 2k}.

A CONSTRUCTIVE LOOK AT THE REAL NUMBER LINE

Let Xk be the 2kth term of the regular sequence defining xV(k); then - Xv(k)I < 1/2k. So for all m and n we have

Ix.- -

1x4 - XV(•m• + IXV(,) - xVwfl + IXv•f < 1/2m + (1/2m + 1/2n)+ 1/2n = 1/m + 1/n.

x71 !5

51

1x7

- x&j

Hence x- = (x,-)n=l is a real number; so Ix- - xn-1 < 1/n for each n. Moreover, if n > v(k), then Ix. - xJl

- Ix- - x,7I + Ix, - XVWJ)I + IXv(,) - Xnl

5 1/n + 1/2n + 1/2k < 1/3k + 1/6k + 1/2k = 1/k. Thus (x,) converges to the real number x-.

U

It is worth noting that the above proof contains two applications of the principle of countable choice: the first occurs when we construct the sequence (Nk), the second when we construct the sequence (x4) by treating the sequence (xJ)of real numbers as a sequence of regular sequences of rational numbers. The completeness of R has many applications, for example in the justification of various tests for the convergence of infinite series. Recall that for each sequence (x,)-,=l of real numbers the real number Sk A,=,X• is called the kth partial sum of the series Y,-=lxn. (Formally, we define the infinite series Y7-,*=x, to be the sequence (s1 , s2 ... ) of its partial sums.) The sum of that series is the limit s of the sequence (sn) of partial sums, if that limit exists; in that case, we say that the series converges to its sum, and we write 1,=,x,, = s. It might be thought that convergence tests for infinite series would not play a role in constructive analysis. For is not the purpose of such tests to enable us to prove the convergence of a series without calculating its sum; whereas to assert constructively that a series converges is, at least implicitly, to provide both an algorithm for calculating its sum and an algorithm that gives the rate at which the partial sums approach the sum? In fact, there is no conflict here: the proof of a constructive convergence test embodies algorithms which, if applied to an infinite series of the appropriate type, calculate the sum of the series and provide the rate of convergence to that sum.

52

DOUGLAS S. BRIDGES

The comparison test for convergence of a series says that (5.2)

If

-,.Iy. is a convergent series of nonnegative terms, and if

IxýI - n > Nk. Then for all such m and n, =xj- X=x _ x j/k, < so the sequence of partial sums of X•=•xn is a Cauchy sequence. Since IR is complete, the desired result follows. 0 More sophisticated convergence tests can also be justified using the completeness of R; see Bishop and Bridges (1985, Ch. 2, Section 3). 6.

OPEN AND CLOSED SETS

Let S be a subset of R. For each set X with S C X C R, the closure of S in X is the set of all points of X that are limits of sequences of numbers in S. The closure of S in R is noted by either S- or (S)-; thus the closure of S in X is s- n X. We say that S is closed in X, or a closed subset of X, if S = s- n X - that is, if all the limits of sequences of numbers in S belong to S. On the other hand, we say that S is open in X, or an open subset of X, if for each s E S there exists r > 0 such that x E S whenever x E X and Ix - sI < r. If the identity of the set X is clear from the context, then, for example, we shall often say that a subset of X is 'closed', rather than 'closed in X'. Examples of open subsets of R are provided by the open intervals, which are sets of the following forms, where a, b are real numbers with a
{x = R {x E R

a <x < b}

{x E

x < b}

a <x}

A CONSTRUCTIVE LOOK AT THE REAL NUMBER LINE

53

Examples of closed sets are the closed intervals, which are sets of the following forms, where a, b are real numbers with a < b: [a, b] [a, (-,oo b] 00)

{x r= R I{x E R {x E R

a • x • b}

a • x} x < b}

By convention, R is considered as both an open interval and a closed interval. For completeness, we also mention here the remaining types of interval: half open on the left half open on the right

(a,b] [a,b)

{x e R: a < x < b}, {x E R a 5 x < b}.

The last two intervals are neither open nor closed in R. In, for example, the interval (a, b), the real number a is called the left end point, and b the right end point, of the interval. Intervals of the form [a, b], (a, b), [a, b), or (a, b], where a, b e R• and a < b, are said to have positive length b - a. If S is open in X, then the complement of S in X, X~ S= {x E X: Vs E S(x # s)}, is closed in X. To see this, consider a sequence (xn) in X - S converging to a limit x. in X. Given s in S, choose r > 0 such that x E S whenever x E X and Ix - sl < r. If Ix- - sl < r, then for all sufficiently large n, Ix, - sl < r and therefore xn E S - a contradiction; so Ix_ - sl > r and therefore x. # s. As s E S is arbitrary, it follows that X- E X - S; whence X - S is closed in X. Classically, S is open in X if and only if X - S is closed in X. Things are rather different constructively: (6.1)

The statement if R - S is closed in R, then S is open in R

entails LPO. Proof. Let S a (-1, 0) U {01 U (0, 1) C R•. It is easy to see that R - S equals (-, -1] U [1, -o), which is closed in R. On the other hand, if S is open in R, then there exists r > 0 such

54

DOUGLAS S. BRIDGES

that x E S whenever lxi < r. It follows that

Vx E R (lxi 0), from which we can derive LPO (cf. (4.5) above).

U

Having first introduced the notion of an open set, some classical authors define a set S to be closed in R if and only if R - S is open. The following result reveals the constructive limitations of such an approach. (6.2) Each of the following statements entails LPO: (i) For each closed subset S of R, R - S is open. (ii) For each subset S of R such that R - S is open, S is closed. Proof. Let (an) be a binary sequence with at most one term equal to 1, and for each n define Sn = {0} = [0, 1-2-n]

ifan = 0, if an= 1.

Then S - (U-.,Sn)- is closed in R; S = {0} if an = 0 for all n; and S = SN if a, = 1. For each x in S we have either x < 1 or x > 0. In the latter case, there exists s E U=,Sln such that s > 0; choosing N so that s E SN, we see that aN = 1 and S = SN; whence x < 1-2-N. Thus in either case, I re R - S. Now suppose that R S is open in R; then there exists r such that 0 < r < 1 and 1 - r E R - S. Choosing a positive integer k such that 2 -k < 1 - r, we see that an = 0 for all n > k; by testing a, . . . , ak, we can now show that either an = 0 for all n, or else an = 1 for some n < k. Thus statement (i) entails LPO. To deal with statement (ii), let S-

{0} U (0,

c).

Then x c= R

-

S € • x 0 and x < 0 Sx < 0,

so R - S equals {x E RR:x < 0}, which is certainly open in R. Now consider any real number x - (xn) ->0, and for each n define yn a xn + 1/n. Then (Yn) is a sequence of numbers in S converging to x. If S is closed in R, then x r S, so either x = 0 or x > 0. Thus (ii) entails

A CONSTRUCTIVE LOOK AT THE REAL NUMBER LINE

VxE R(x>O

55

=ýx=Oorx>0),

from which we can derive LPO.

0

The reader is invited to prove that the union of a sequence of open sets is open; the intersection of a finitely enumerable set of open sets is open; and the intersection of a sequence of closed sets is closed. However, without the help of classical logic, he cannot prove that the union of two closed sets is closed: (6.3) The statement The union of two closed subsets of R is closed entails LLPO. Proof. Assume that the statement in question holds, and define closed subsets A, B of RI as follows: A I{x • : x 01, B -{x i: x _01. Consider any binary sequence (an) with at most one term equal to 1, and define a sequence (x,) of rational numbers in A U B by xn = (-1)k/k = 0

if k < n and ak = 1, otherwise.

Then x - (x,) is a regular sequence - that is, a real number. Since Ix - x.1 < 1/n for each n, the sequence (xn) converges to x, which therefore belongs to the closed set A U B. If x s A, then an = 0 for all even n; ifx e B, then an = 0 for all odd n. 7.

SUPREMA AND INFIMA

Let X be a nonvoid subset of R. A real number b is an upper bound of X if x < b for all x in X; the supremum, or least upper bound, of X if it is an upper bound of X, and if, for each E > 0, there exists x in X such that x > b - F. If X has an upper bound, then we say that X is bounded above. It is readily shown that the supremum of X, if it exists, is unique; we denote it by sup X.

56

DOUGLAS S. BRIDGES

On the other hand, b is a lower bound of X if b !5x for all x in X; the infimum, or greatest lower bound, of X if it is a lower bound of X, and if, for each E > 0, there exists x in X such that x < b + F. If X has a lower bound, then we say that X is bounded below. The infimum of X, if it exists, is unique, and is written inf X. We say that X is bounded if it is both bounded above and bounded below. A cornerstone of classical analysis is the least-upper-bound principle, which says that every nonvoid subset of R that is bounded above has a supremum. (7.1)

The statement every nonvoid subset of {0, 1} has a least upper bound

entails LPO. Proof. Let (an) be a binary sequence with a, = 0, and let S {a : n E 'J+}.Then S is a nonvoid subset of {0, 1}. If S has a least upper bound s, then either s > 0, in which case, computing N such that aN > s/ 2 , we have a, = 1; or else s < 1, and a. = 0 for all n. M Fortunately, there is an extremely useful constructive substitute for the least-upper-bound principle. (7.2) Let S be a nonvoid set of real numbers that is bounded above. Then sup S exists if and only if for all a, P3 in R with a < j5, either [3 is an upper bound of S or else there exists x in S with x > ax. Proof. If M - sup S exists and x < f3, then either M < j3 or M > a. In the former case, P is an upper bound of S; in the latter, we can find x in S with x > M - (M - x) = a. Assume, conversely, that the stated condition holds. Choose an upper bound u, of S, and e > 0, so that [u0 - e, u0] n S is nonvoid. With F. = (3/4)e, construct a sequence (u.) of upper bounds of S inductively, such that for each n, (i) [u, - F,, u.] n S is nonvoid; (ii) either un,,I = un - EnI4 or u,,, = u,. To do so, assume that u0 . . . . . . u have been constructed with the appropriate properties. Then either u. - F_,/4 is an upper bound of S, or

A CONSTRUCTIVE LOOK AT THE REAL NUMBER LINE

57

else there exists ý in S with 4 > u,- F,,,. In the first case, set Un+u, - e,/4; in the second, set u,,÷- un. This completes the inductive con-

struction. By (ii), (*)

1u.+ 1 -

unI < eJ4 for each n; so whenever m > n we have IT==n 1IUk+l

Urn - Un I

-

Ukl

'k=4E4

< 17=.•-/4 =4¼,17=.(3/) =

(3/4)n, = F.

So (u,) is a Cauchy sequence and therefore, by (5.1), converges to a limit u- in R. If u- < x for some x in S, then un < x for all sufficiently large n; since this contradicts the fact that un is an upper bound of S, we conclude that u- is an upper bound of S. On the other hand, by (*) above, Urn> un - £- whenever m > n; so u- > uý - En. Therefore, by (i), [un - n, u_] f S = [un - En, uj] n S is nonvoid for each n. It now 0 follows that u- is the supremum of S. In this last proof there is an application of the principle of dependent choice: for a given value of n we may have both un - FQ4 an upper bound of S and 4 >

Un - F.n+

for some 4, and the possible choices

for u,,, depend on u,. However, it is possible to modify the proof so that only countable choice is involved (Bridges and Richman, 1985, Ch. 2, Problem 1). The criterion of the constructive least-upper-bound principle can be applied in more restricted forms in certain contexts: for example, if S is a set of nonnegative numbers, then to prove that sup S exists it will suffice to show that for all nonnegative numbers x, 13with cc < 13, either P3 is an upper bound of S or else there exists x in S with x > a; it will even suffice to show that for all such a and 0, sup S exists or 0 is an upper bound of S or there exists x in S with x > a. It is almost trivial to deduce from (7.2) the corresponding result about infima. Let S be a nonvoid set of real numbers that is bounded below. P in R with x < P, either ax 0 is a lower bound of S or else there exists x in S with x < P. (7.3)

Then inf S exists if and only if for all a,

58

DOUGLAS S. BRIDGES

If x, y E R, then sup{x, y} exists and equals max{x, y}, and inf{x, y} equals min{x, y}. More generally, the supremum (and, similarly, the infimum) of a finitely enumerable subset {x . . . . . . x.} of R exists, since for all real numbers x, P3 with a < P3,either xk < P for all k or else Xk > x for some k. Now let X be a subset of R, and S a nonvoid subset of X. We say that S is located in X, or a located subset of X, if the distance dist(x, S) = inf{Ix - yJ : y E S1 from x to S exists for each x in X; if the identity of the set X is clearly understood, we often simply say that S is 'located', rather than 'located in X'. For example, a closed interval [a, b] is located in R: in fact, as the reader may verify, dist(x, [a, b]) = max{0, a - x, x - bI for all x in R. Using (7.3), we now derive a criterion for locatedness. (7.4) A nonvoid subset S of R is located if and only if for each real number x, and all real numbers ax, P with 0 < ax < P, either (x - 1, x + P) intersects S or (x - a, x + a.) is disjoint from S. Proof. If S is located, x E R, and 0 < x < P, then either dist(x, S) < 13or dist(x, S) > ax. In the first case, there exists s in S such that Ix - sl < 13,so s ( (x - 13,x + P); in the second, (x - ax, x + a) is disjoint from S. Conversely, if the stated condition holds, then for all real numbers a, 13with 0 < a < 13, either Ix -YI > a for all yin S or else Ix-sl < 1 for some s in S. So either ax is a lower bound for D = {Ix y E SI, or else there exists 8 in D such that 8 < 13. It follows from (7.3) that S is located. 0 8.

CANTOR'S THEOREM; BINARY EXPANSIONS

Our first aim in this section is to prove that the real line is uncountable. To do so, we need a simple lemma. (8.1) Let I =_ [0, 1],andfori= 1, 2, 3, let Ji= [(i- 1)/3, i/3]. Then for each real number x, either dist(x, J1) > 0 or dist(x, J3 ) > 0.

A CONSTRUCTIVE LOOK AT THE REAL NUMBER LINE

59

Proof. Either x > 1/3, in which case dist(x, J,) = x - 1/3 > 0; or else x < 2/3, in which case dist(x, J3) = 2/3 - x > 0. 0 In view of (4.6) we cannot prove constructively that I = J, U J 2 U Since, classically, a number between 0 and 1 belongs to JI if it has a ternary expansion whose fractional part has first digit i - 1, it appears that there may be a problem with the construction of expansions of real numbers. We shall return to this problem shortly. We have already remarked upon the misconception that the constructive real line R is not complete. Another common misconception is that R must be countable. It is true that the recursive real line is classically countable; but within the framework of BISH, RUSS, or INT the corresponding real number line is uncountable. To justify this statement, since RUSS and INT are formally consistent with BISH it will suffice to prove the following version of Cantor'stheorem within BISH. J 3.

(8.2) If (an) is a sequence of real numbers, then in any closed interval of R with positive length there exists a real number x such that x # an for each n. Proof. Let I0 be a closed interval with positive length. We construct inductively a sequence Io, 1, .... of closed intervals with positive length such that for each n > 1, (i) In C In_, and JII, = -Ll n (ii) dist(ak, In) > 0 for k = 1, . . . , n. Indeed, having constructed I0. ... 1 _nwith the relevant properties, we need only apply (8.1) to produce a closed interval In, with positive length, such that (i) obtains and such that dist(a., I.) > 0. Clearly, since also dist(ak, I,-,) > 0 for k = 1..... n- 1, property (ii) holds. This completes the inductive construction of (In). For each n > I let xn be the midpoint of In. It follows from (i) that IXm

-xnI-<

JInJ = (1/3)n1101

whenever m > n > 1. So (xn) is a Cauchy sequence in 10; let x be its limit in R. For each n, xm E In whenever m > n; whence x E In, as In is closed. It follows from (ii) that for each n > 1, Ix - anI > dist(a., In) > 0, and therefore x # an. E The above proof is essentially Cantor's diagonal argument applied to the ternary expansions of the terms a.. However, as we suggested

60

DOUGLAS S. BRIDGES

above, and as we are about to demonstrate, ternary (and other) expansions of real numbers cannot always be constructed. For simplicity, we shall focus our attention on binary expansions. A binary expansion of a nonnegative real number x is a binary sequence (b,)-=0 such that x = Y•= 0b=2-; following convention, we then write x = bo'blb 2 . . . . Note that for any binary sequence (b.)-= 0 the series YX= 0bb2- converges, by comparison with I-= 02-"; see (5.2).

A typical classical proof that a real number x has a binary expansion involves an interval-halving argument of the following sort. For simplicity, take 0 5 x < 1 and set x0 - 0, I0 = [0, 1]. Having found the binary digits b0..... b. and the closed intervals I0..... In =_[ps, qJ], write m =-2(p, + qn). If x E [p., M), set b,+1 - 0 and In+, a [pa, m]; if x e [m, q.], set b.+, = 1 and I,+, -= [m, qJ]. The completes the inductive 'construction' of a binary expansion bo.blb 2 . . . of x.

Such an argument will not work in the constructive framework, since, for example, we cannot there decide, for an arbitrary real number x, whether x < 1/2 or x > 1/2. It is therefore no surprise that (8.3) The statement each real number x has a binary expansion entails LLPO.

Proof. Let (aJ)be an arbitrary binary sequence with at most one term equal to 1, and define a real number x

x, = 1 = 1 + (-1)k/k

Note that if ak

= I

-(xn)

=. as follows:

ifak =Oforallk
for an odd value of k, then x < 1; whereas if ak

= 1

for an even value of k, then x > 1. Suppose that x has a binary expansion bo.blb 2 . . . . If b, > 0, then -- (x < 1), so ak = 0 for all odd k; if 0 b 0 < 1, then -,(x > 1), so a, = 0 for all even k. Perhaps the reader is asking himself why we do not concentrate our attention on those constructive real numbers for which a binary expansion can be computed. The reason for our choice of a notion of real number that includes objects that do not have computable binary expansions lies in the following result.

A CONSTRUCTIVE LOOK AT THE REAL NUMBER LINE

61

(8.4) The statement for all real numbers x, y with binary expansions, the sum x + y has a binary expansion is equivalent to LLPO. Proof. Consider any binary sequence (an) with a, = 0 and at most one term equal to 1, and define a new binary sequence (b.) as follows. Ifak= 0 for all k < n, set b--0; ifn is odd and an = 1, set b. 0, bn֥ -0, and bk =- I for all k > n + 2; if n is even and an = 1, set bk 1 for all k > n. Let x be the real number with binary expansion 0-b~b2 . . . On the other hand, let y be the real number with binary expansion 0.cc 2 .... where c, = I and for all n > 2, c, = 1 = 0 If an

=

ifak=0forallk<notherwise.

1,

0 for all n, then x + y = 1; if a,= I for an odd N, then

x

+ y =

IN=

I -

-n 2-+I-f=N112

2 -N-1;

and if aN = 1 for an even N, then

, 2 -n +

x+y=

1nX=N2-

=

I +

2-N.

Suppose x + y has a binary expansion do.dfd2

.

.

.

. If do > 0, then

x + y > 1, so a. = 0 for all odd n; whereas if d0 < 1, then x + y

s,

an = 0 for all even n. Thus we can prove LLPO. We leave the reader to prove the converse.

N

9.

TOTALLY BOUNDED SETS

Possibly the most important subsets of R• for which suprema and infima can be computed are those that can be approximated arbitrarily closely by finite sets. Let S be a set of real numbers, and E > 0. A nonvoid subset E of S is called an e-approximation to S if for each x in S there exists y in E such that Ix - yI < F. S is said to be totally bounded if for each F > 0 there exists a finite E-approximation to S. Some authors allow e-approximations, and therefore totally bounded sets, to be empty. To avoid complicating the relation between total boundedness and locatedness, we shall only allow nonvoid E-approximations.

62

DOUGLAS S. BRIDGES

It is an exercise to show that S is totally bounded if and only if for each e > 0 there exists a finitely enumerable E-approximation to S (Bridges and Richman, 1987, Ch. 2, (4.1)). (9.1) Let a, b be real numbers with a • b. Then the closed interval [a, b] is totally bounded; and if a < b, the open interval (a, b) is totally bounded. Moreover, if a < b, N is a positive integer, and ai =_ N), then {a . . . . . . aN-1) is a a + i(b - a)/N for each i E {0 ..... finite (b - a)/N-approximation to either interval. Proof. For each e > 0, either b - a < F, in which case {a} is an E-approximation to [a, b]; or else a < b. Thus it will suffice to consider the case a < b. With N and ai as above, let T= {ai: i= 1,....

N-

1},

which is finite. Consider any x in [a, b]. Either x < a2 , in which case Ix - a1l < (b - a)/N; or, as we assume, x > a,.Then either x < a 3, in which case Ix - a2l < (b - a)/N; or x > a 2. Carrying on in this way, we show that Ix - ail < (b - a)/N for some i with I < i < N- 1; whence T is a finitely enumerable (b - a)/N-approximation to both [a, b] and N (a, b). (9.2) If S is a totally bounded subset of R, then sup S and inf S exist. x3 Proof. Let ax, be real numbers with a < P3,and set - ( -)/3. Let {x . . . . . . x,} be a finite £-approximation to S, and choose N in ..... n} .1such that XN > sup{x . . . . . x,} - F. Either ax < x, or XN < x + F. In the latter case, if s is any point of S and i is chosen so e , then that Is - x,I

P

s

xi + e <xN + 2£ < ca + 3F=

f,

so that 3 is an upper bound of S. Thus sup S exists, by (7.2). By conE IR : -x E S), we see that inf S = -sup T exists. sidering T=- {xx (9.3) A totally bounded subset of R is bounded and located. Proof. If S C R is a totally bounded set, then, choosing a finite 1-approximation F to S, we have IsI -<1 + sup F for all s in S; whence S is bounded. On the other hand, given £ > 0, let {x . . . . . . x,} be a finite £-approximation to S, and consider any x in lR. For each s in S there

A CONSTRUCTIVE LOOK AT THE REAL NUMBER LINE

63

exists i such that Is - xi < ; then

IIx - sl - Ix- xill -
-

x,1..... Ix - xI} is an e-approximation to

{Ix

T

-s

:s E S}.

As F is arbitrary, T is totally bounded; whence dist(x, S) = inf T exists, by (9.2). 0 (9.4) If S is a totally bounded subset of R, and A a located subset of S, then A is totally bounded. Proof. Given £ > 0, let {xl..... x.1 be an E/3-approximation to S. For each j, choose aj in A such that Ixj - ajl < dist(xj, A) + E/3. Consider any point y of A. Choosing i such that lY - x11 < e/3, we see that dist(xi, A) < Ix, - yl - 0 such that S C [-c, c]; so, by (9.1) and (9.4), S is totally bounded. Reference to (9.3) completes the proof. E It is convenient to introduce here an important totally bounded subset of R that will reappear in our discussion of Specker's theorem, in Section 11 below. The Cantor set is the subset C

{Y= 1c3-

C c:

{0, 2} for each n

of the unit interval [0,1]. It is fairly easy to show that C is closed in R, and therefore complete; and that if a and b are two numbers in C which differ in the mth place of their ternary expansions, then la - bi > 3-'. Also, for each positive integer n the finite set Yk =ICk3 k: Ck

10, 2) for 1
64

DOUGLAS S. BRIDGES

is a 3-"-approximation to C: for if x =-1=,ck3-k is any point of C, then Ix

7=f+l1Ck3-k = -lCk3'kI

X

-

< 2X7.=ý+13-k

=

3-n.

So C is totally bounded, and therefore located. 10.

METRIC COMPLEMENTS

Let X be a subset of R, and S a subset of X. The metric complement of S in X is the set of those points x of X that are bounded away from S. in the sense that there exists x > 0 such that Ix - sI > cx for all s in S; the metric complement is written X - S, or, if the identity of X is clear, simply -S. Note that X - S is open in X, and that if S is located in X, then

X

S = {x E X: dist(x, S) > 01.

-

Let S be located in R, and x E R; there exists a point y of S-, depending on x, such that if x # y, then x E R - S: (10.1) Let S be a located subset of R, and x a real number. There exists a number y in S- such that for each positive integer n, if Ix - Yj > 4-"'l, then dist(x, S) > 4-'. Proof. Since for each n we have either dist(x, S) < 5.4-"- or dist(x, S) > 4-f, we can use countable choice to construct a (perforce increasing) binary sequence (k.) such that X. = 0 =>' dist(x, S) < 5.4 Xý = 1 * dist(x, S) > 4-". Let s be any point of S. If k, = 1, set X1 s; if kn = 0, choose x, in S such that Ix - x.1 < 5.4"-'; if n > 1 and X., = 1, set x, = x,-,. Then (x,) is a Cauchy sequence in R. In fact, whenever m > n we have Ix,. - xnI < 10.4"-': for if Xm = 0, then X,, = 0 and IX.

-

XnI

IX -

X.i

+

IX -

Xn

< 5.4-m-1 + 5.4-"-n < 10.4" -1; if X,, = 1, then X.m = 1 and x. = x,; if Xm = 1 and ., = 0, then there exists k such that n < k < m and ,k = 1 - X,-,, so that, by the first case we considered, IXm - X[ = IXk_, - xnI <

10.4--

A CONSTRUCTIVE LOOK AT THE REAL NUMBER LINE

65

By (5.1), (x,) converges to a limit y in S- such that ly - x.1 < 10.4-n-I for all n. Now consider any positive integer n. Supposing that Ix - Yj > 4-+1, we have Ix - xj > Ix - y[ - lY - xn[ > 4`1 - 10.4-n-1 = 6.4"; so X•,cannot equal 0, and therefore k. = 1. Hence dist(x, S) > 4 -.

N

The above result is trivial to prove classically: if dist(x, S) < 4-, choose y in S so that Ix - yI < 4-nl1; if dist(x, S) > 4'", let y be any point of S. Results like (10.1), which have almost no classical content but require some ingenuity to prove constructively, are often both interesting and useful in the constructive context. An amusing result of this type is found in Bridges and Richman (1987, Ch. 2, (6.6)); another, not particularly useful, example is the following partial converse to (10.1). (10.2) Let S be a nonvoid bounded subset of R with the following property: for each x E R and each positive integer n, there exists a number y in S- such that if Ix - yI > 4-"÷l, then Ix - sI > 4-' for all s E S. Then S is totally bounded and hence located. Proof. Choose c > 0 such that S C [-c, c]. Given 6 > 0, compute a positive integer N such that 2(4-N) + 4'N+2 <

F.

Let {x . . . . . 1Xm} be a 4-N-approximation to [-c, c], i {1..... ml choose y1 in S- such that if 1xi - yI 1x, - sl > 4 N for all s E S; choose also s5 E S such that Construct numbers X...... 'm in {0, 1} such that

and for each > 4 -N+1, then y1 - s1i < 4-N.

= 0 • Ixi - yI < 4'2, ki = I = Ix1 - yj > 4-N.l Let x E S, and choose i such that Ix - x1i < 4N. If X, = 1, then, by our choice of yi, we have Ix - xil > 4-N, a contradiction. So X, = 0 and therefore 1xi - yjI < 4'N+2. Hence ki

Ix - sl -• Ix - xil + lxi - y11 + lyi - sil < 4N + 4-N+2 + 4'N < E. Thus {si : k, = 0) (which, as we have shown in passing, is nonvoid) is a finitely enumerable 6-approximation to S. As e > 0 is arbitrary, the desired conclusion follows. N

66

DOUGLAS S. BRIDGES

If S C X C R, then the metric complement X - S is a subset of the complement X - S. Classically, if S is closed in X, then X - S = X - S; as is so often the case, things are not so straightforward constructively. (10.3) IfS is a closed located subset of R, then R - S = R - S. Proof. Since R - S C R - S, the desired result follows from the fact that, by (10.1), for each x in R - S there exists y in S such that if x # y, then xE R- S. (10.4)

The statement if S is a nonvoid closed subset of R, then R - S = R - S

entails LPO. Proof. Let (an) be a binary sequence with at most one term equal to 1, and define Sn = {0} = [0, 1 -2-]

ifa, =0, if an= 1.

Let S be the nonvoid closed subset (U-=, Sn)- of R. We saw in the proof of (6.2) that if R - S is open in R, then either a. = 0 for all n or else there exists n with a. = 1. Since R - S is open if it equal R - S, the result follows. 0 We see from (10.3) and (10.4) that (10.5)

The statement every nonvoid closed subset of R is located

entails LPO.

0

Finally, note that in (10.3) we cannot replace R by an arbitrary subset X, even if we add the condition that S be located in R: (10.6)

The statement if X C R and S is a closed subset of X that is located in R, then X- S = X- S

is equivalent to Markov's principle. Proof. With a any real number such that -(a < 0), define S

A CONSTRUCTIVE LOOK AT THE REAL NUMBER LINE

67

(-1, 0) and X =-S U {al. Then S is located in R. Let (sn) be a sequence in S converging to a limit s- in X, and note that s- < 0. Then either s- E S or s_ = a; the latter case is ruled out, as it entails a < 0. Thus S is closed in X. It is clear that a E X- S, and that if a E X - S, then a > dist(a, S) > 0. Thus the statement in question entails Vx E R (-,(x < 0) = x > 0), which, in turn, entails Markov's principle. Conversely, suppose Markov's principle holds. Let X C R, let S be a closed subset of X that is located in R, and let x be any point of X - S. By (10.1), there exists y in the closure S- of S in R such that if x # y, then dist(x, S) > 0. If x = y, then y E s- n X = s, so x E s, which is absurd; hence -(x = y). Therefore by Markov's principle, x # y; so dist(x, S) >0 and xE X - S. U 11.

SPECKER'S THEOREM

In this section we discuss a famous theorem in Specker (1949) which provides a very strong 'recursive counterexample' to the classical theorem that a bounded increasing sequence of real numbers converges to its supremum. (The latter theorem is, of course, one expression of the least-upper-bound principle.) Not wishing to become embedded in the morass of technical detail that characterises many presentations of recursive function theory, for the details of which we refer the reader to Rogers (1967), we now state the constructive form of the Church-Markov-Turing thesis: CMT

Every partialfunction from N to N with countable domain is recursive.

What we actually require for our proof of Specker's theorem is the following consequence of that thesis:

CPF

ml,

There is an enumeration9 (po, (P2 .... of the set of all partial functions from N to N with countable domains.

Referring to (2.4) above, we fix an effective enumeration (P, qp, . of the set of partial functions from N to N, and for each natural number n, an increasing sequence (D,(k))7=0 of finite subsets of N whose union is the domain of (p.. The reader may find it helpful to consider the intuitive model in which D,(k) consists of those natural numbers x < k

68

DOUGLAS S. BRIDGES

such that a computer program for the calculation of (p, computes (p,(k) in at most k + 1 steps. (For further information about CMT, CPF, and recursive constructive mathematics, see Chapter 3 of Bridges and Richman (1987).) We can now prove Specker's theorem. (11.1) Under the Church-Markov-Turing thesis, there exists an increasing sequence (r.) of rational numbers in the Cantor set C with the following property: for each real number x there exist a positive integer N and a positive number 5 such that Ix - rnI > 6 whenever n > N. Proof. For each positive integer n let r, Y =,=s,(m)3-, where for m < n, if m E Dm(n) and (pm(m) = 0, otherwise.

s,(m) = 2 = 0

Then r, is rational and belongs to C. Also, since Dm(n) C Dm(n + 1), if s,(m) = 2 then s,+,(m) = 2; so r, _ m so large that m e Dm(N). Then for all n > N, p(p(m) = 2 - s,(m), by definition of s.; so x and r, differ by 2 in the mth place of their ternary expansions, and therefore Ix - rjI > 3-m = 8. This completes the proof when x E C. Now consider any real number x. By (10.1), there exists c in C such that for each positive integer n, if Ix - cl > 4-nf', then dist(x, C) > 4-. By the first part of the proof, there exist positive integers M and N such that Ic - r.I > 3.4--+' whenever n > N. Let 8 -4-.M Either Ix - cl < 2 .4 -M+', in which case Ix - rnI >

Ic

- rnI - Ix -

cl Ž-4-M+'

> 5

whenever n > N; or else Ix - cl > 4 -M+. In the latter case, Ix - r.I > dist(x, C) > 4-M = 8 for all n.

a

A CONSTRUCTIVE LOOK AT THE REAL NUMBER LINE

69

We can modify the sequence (rn) in Specker's theorem to ensure that it is strictly increasing. For, with (r,) as in the conclusion of (11.1) and n 1, we can construct inductively a strictly increasing sequence (nJ7)7. of positive integers and a sequence ( 8 )7=1 of a positive numbers such that for each k, Irj - r J > 8, whenever j > nk; we then need only replace rk by rflk. A strictly increasing sequence (rn) of real numbers with the property described in the conclusion of Specker's theorem is called a Specker sequence. Such sequences are the basis of many strong 'recursive counterexamples' to classical theorems. For example, Figure 1 below illustrates the construction of a function f from [0, 1] onto [0, 1] with the following properties: (i) f is pointwise continuous: that is, for each x in [0, 1] and each e > 0, there exists 8 > 0 such that if y E [0, 1] and Ix - yI < 8, then If(x) -f(y)l < E. (ii) f is not uniformly continuous: to be precise, there exist sequences (x,), (y.) of real numbers between 0 and 1 such that for each n, < l/n,f(x,) = 0, and f(y.) = 1. Ix. - y,,I The existence of such a function sharply contrasts with the classical situation, in which every pointwise continuous function f from [0, 1] to R is uniformly continuous (in other words, in (i) the constant 8 can be chosen to depend only on E, and not on x).l°

I

0

Fig. 1.

<1/2

<1/3

<1/4

A pointwise continuous map of [0, 1] onto [0, 1] that is not uniformly

continuous.

70

DOUGLAS S. BRIDGES

Another recursive counterexample to a classical theorem is illustrated in Figure 2. This time, we have a pointwise continuous function f from [0, 1] onto (0, 1]; thus inf{f(x) : x E [0, 1]} = 0, but f(x) > 0 for all x in [0, 1]. (In classical mathematics, a pointwise continuous function from [0, 1] onto (0, 1] must attain its infimum at some point; whence that infimum is positive.) Even more remarkably, we can construct a recursive example of uniformly continuous function f from [0, 1] onto (0, 1]. For the details of such examples, see Bridges and Richman (1987, Ch. 3, Section 3, and Ch. 6).

1

1/2 1/3

114

I

I/2 1/3 1/4

0-

rI

I

r2

r3

Fig. 2. A pointwise continuous map of [0, 1] onto (0, 1].

12.

LOCATED SETS

Our aim in this section is to characterise the located subsets of the line. In doing so, we modify and extend the work in Mandelkern (1981). (12.1) If S is located in X, then S- n X is also located in X. Proof. Given x in X and y in S- n X, suppose that Ix - yI < r =-dist(x, S). Choose s in S such that ly - sl < r - Ix - yl. Then Ix

-

sl • lx

-

yj + ly

-

sl < r,

a contradiction. So Ix - YŽ> r for all y in S-

n X. Since S C S- n x,

A CONSTRUCTIVE LOOK AT THE REAL NUMBER LINE

for each c > 0 there exists z in S- n X such that Ix - zI < r + dist(x, S- n X) exists and equals r.

F.

71

Hence E

Let S C X C R. We say that S is dense in X if for each x in X and each e > 0 there exists s in S such that Ix - sl < e. For example, Q is dense in R. (12.2) If S C R is located, then S U -S is dense in R. Proof. For each x in R and each E > 0, either dist(x, S) > 0 or dist(x, S) < F. In the former case, x E -S; in the latter, there exists s in S such that Ix - sl < e. 0 The reader is invited to ponder the question: if S is a nonvoid subset of R, is S U -S dense in R? The following result can also be proved using the observation that if dist(x, T) exists for all x in a dense subset of R, then T is located in

R. (12.3) Let S be a located subset of R, T a real number, and T S n (--oo, t]; suppose that r = sup T Then T is located in R; in fact, dist(x, T) = max {x - t, dist(x, S)} for all x in R. Proof. Given x E R, let r -max{x - C,dist(x, S)J. If y E T and Ix - yj < r, then as Ix - YI > dist(x, S), we have x - y < Ix - yI < r = x- -r; so t < y, a contradiction. Thus Ix - YI > r for all y E T. On the other hand, for each F > 0 we have Ix - tl < s or x > t or x < t. In the first case, Ix - cl < r + e; in the second, dist(x, S) < x - t (as '" = sup T and T C S), so x - c = r < r + E. In the third, r = dist(x, S) and we can choose y C S such that Ix - YI < r + d/2; if y < ', then y E T; if y > t - e/2, then

Ix - 't = "t - x < t - y + Ix-yl < e/2 + r + e/2 = r + 6. So in all cases there exists y in T such that Ix - yI < r + arbitrary, it follows that dist(x, T) exists and equals r.

F.

As e> 0 is N

Note that we cannot remove from (12.3) the restriction that t sup T:

=

72 (12.4)

DOUGLAS S. BRIDGES

The statement if S is a located subset of R such that T- S= s nonvoid, then T is located in R

(--, 1) is

entails LPO. Proof. For any real number r with 0 tc < 1/2, let S be the finitely enumerable, and hence located, subset {0, 1 - c} of R. Then TS n (--,•, 1) is nonvoid. If 8 - dist(1, T) exists, then either 8 > 1/2 or 5 < 1. In the first case, if 'r > 0 then we obtain the contradiction 8 < 1 - (1 - t) < 1/2; so ct< 0 and therefore "r = 0. In the case 8 < 1, > 0. Thus the there exists 4 in S such that 0 < 4 < 1; sot = I1 statement in question entails Vx E R (x > 0 = x> 0 or x

=

0),

from which we can derive LPO.

U

(12.5) Let S C R be located, and a, b points of S such that a • b. Then T = S n [a, b] is totally bounded. Proof. For each c > 0, either b - a < s, in which case {a} is an e-approximation to T; or, as we assume, a < b. Compute a positive integer N > 3(b - a)/2•, 2N, write xi = a + i(b - a)/2N. By (9.1), and for i = 0, 1..... {X0 . X2N} is a (b - a)/2N-approximation to [a, b]. Since a, b e S, X.2. in (0, 1} such that we can construct integers ), .. -. ,2N

S=

k

=2N

=

,2N = 0

and such that for each i, ,i = 0 * dist(x1 , S) < (b - a)/N, ki = I = dist(x,, S) > (b - a)/2N. {2..... 2N - 21 Let ao= a, =--a, a2N-1 = a2N -b, and for each i with X, = 0, choose a, E S such that Ix, - ail < (b - a)/N. Note that for such i, as x 2 < xi! X2N-2, we have a < a, < b; so a, E T. Given x in T, choose i such that Ix - xl < (b - a)/2N. Then dist(x,, S) < (b - a)/2N, so X, # I and therefore X, = 0. Hence

Ix - aI - Ix - xil + Ixi - ail < (b - a)/2N + (b - a)IN = 3(b - a)/2N < E.

A CONSTRUCTIVE LOOK AT THE REAL NUMBER LINE

73

Thus {ai :,i = 0} is a finite 8-approximation to T. As E is arbitrary, T is totally bounded. 0 The proof of the following observation is left to the reader: (12.6) If X C R, S C X is located in X, x E X, and 0 < r = dist(x, S), then (i) (x-r,x+ r) n xcx-s; (ii) inf{dist(s,{x - r, x + r)) : S E S = 0. (12.7) Let S C R be located, a E S, and b a point of R - S with b > a. Then A a S n [a, b] is totally bounded. Proof. Let r - dist(b, S) > 0, and note that, by (12.6(i)), a < b - r and A = S n [a, b - r]. Consider any E with 0 < 8 < r. Either a + 8 > b - r, in which case {a} is an E-approximation to A; or, as we assume, a < b - r. By (12.6(ii)), we can construct c E S such that dist(c, {b - r, b + r}) < 8 - min{I,

b - r - a).

By (12.6(i)), either b - r - 8 < c < b - r or c > b + r. In the first case, a < c; so, by (12.5), there exists a finite 8-approximation F to S n [a, c]. Since b - r - 8 < c < b - r, and therefore for each x in A either a 5 x < c or b - r - <x < b - r, it follows that F U {c} is a finitely enumerable 8-approximation to A. On the other hand, if c > b + r, then again a < c and there exists a finite 8-approximation {x . . . . . . xN} to S n [a, c]. In view of (12.6(ii)), we can partition {1 ..... N) into subsets P, Q such that x, < b - r if i E P, and xi > b + r if i E Q. Then, as c < r, it is easily seen that {xi : i e P} is a finite 8-approximation to A. So in all cases, A has finitely enumerable 8-approximations; whence A is totally bounded. N (12.8) Let S C R be located, and b a point of R - S such that T =S n (--, b] is nonvoid. Then r =-sup T exists, T is located in R, and dist(x, T) = max{x - t, dist(x, S)) for each x in R. Proof. Construct a in T. By (12.7), A -S n [a, b] is totally bounded; so sup A exists, by (9.2). Clearly, t =-sup T exists and equals sup A. To complete the proof, we need only refer to (12.3), since T = S n (-o-, T]. U Note that, in contrast to (12.4), under the hypotheses of our last result

74

DOUGLAS S. BRIDGES

we can prove that S n (--0, b) is located: in fact, S n (-o-, b) = S n (-oo, b].. A nonvoid subset S of R is said to be locally totally bounded if for each bounded subset B of S there exists a totally bounded set K such that B C K C S. Since each interval of the form [a, b], where a < b, is totally bounded, R is locally totally bounded. We now come to a characterisation of located subsets of the line. (12.9) The following are equivalent conditions on a nonvoid subset S of R: (i) S is located in R; (ii) for each a e R such that S n (-00, a] is nonvoid, and each e > 0, there exists ý E R such that Ia - 41 < P and S n (_00, •] is located in R; (iii) for each a E R such that S n [a, is nonvoid, and each , > 0, there exists 4 E R such that Ia - 41 < F and S n [•, is located in R; (iv) for all a, b e R such that S n [a, b] is nonvoid, and for each - > 0, there exist 4, T E DRsuch that la - 41 < e, lb - ill < e, and S n [R, 11] is totally bounded; (v) S is locally totally bounded. Proof. If S is located in R, then since S U -S is dense in R it readily follows from (12.3) and (12.8) that (i) implies (ii). It is easy to show, by considering sets of the form {x E R : -x E S}, that (ii) and (iii) are equivalent. Now assume (ii) and consider any x in R. Choose 4 > x such that S n (-o-, 4] is located, and let r = dist(x, S n (-00, 4]). Then choose t > 4 + r such that T S fn (_--, t] is located. For each y in S, either y < t or y > 4 + r. In the first case, y E T and so Ix - yl > dist(x, T); in the second, Ix - yl > r > dist(x, T). As T C S, it follows that dist(x, S) exists and equals dist(x, T). Hence (ii) implies (i). The proof that (iii) implies (i) is similar. To prove that (i) implies (iv), assume (i) and consider any a, b eD R such that S n [a, b] is nonvoid. Given e > 0, construct 4, T1in S U -S such that a - E < 4 < a and b <11 < b + F. We prove that S n [4, 1l1, which is certainly nonvoid, is totally bounded. If 4 and 1i both belong to S, we need only invoke (12.5); if 4 E S and 11 E -S, we use (12.7); if ý e -S and 11 E S, we apply (12.7) to {x E DR: -x E S} n [-1, -J]. This leaves the case where both 4 and 1l belong to -S. In that case, 00)

0)

A CONSTRUCTIVE LOOK AT THE REAL NUMBER LINE

75

choosing s in S n [a, b], we use (12.7) to construct a finite E-approximation F to S n [4, s], and a finite &-approximation G to S n [s, Ili. Then F U G U {s} is a finitely enumerable e-approximation to S n [I, ill. As e is arbitrary, this completes the proof that (i) implies (iv). Next, assume (iv) and let B C S be a bounded set. Choose a, b E R such that S n [a, b] is nonvoid and includes B. Then there exist 4, T1 in R such that 14 - a + 11 < 1/2, [-q - b - 11 < 1/2, and s n [g, 1] is totally bounded. Clearly, 4 < a and Tj > b; so B C S n [4, T1]. Hence (iv) implies (v). To complete the proof, it will suffice to prove that (v) implies (i). To this end, assume (v), fix s in S, and consider any x in R. Let r = Ix - si + 1, and let B be the bounded subset S n [x - r, x + r] of S, which contains s. Choose a totally bounded set T such that B C T C S. For each y in S, either Ix - yI < r, in which case y E B C T and therefore Ix - yj > dist(x, T); or else Ix - YI > Ix - sl > dist(x, T). Hence dist(x, S) exists and equals dist(x, T). 0 We cannot strengthen conditions (ii)-(iv) of Theorem (12.9) by replacing approximation with exactness. Before proving this, we introduce the weak limited principle of omniscience: WLPO

If (an) is a binary sequence, then either Vn(an = 0) or else -Vn(an = 0).

As with LPO and LLPO, there are compelling arguments for believing that WLPO will never be proved constructively. Accepting those arguments, we now show that the natural strengthening of (12.9) is essentially nonconstructive. (12.10) Each of the following statements entails WLPO: (i) If S is a located subset of Rf, then S n (-oo, a] is located whenever it is nonvoid. (ii) If S is a located subset of R, then S n [a, -o) is located whenever it is nonvoid. (iii) If S is a located subset of R, then S n [a, b] is totally bounded whenever it is nonvoid. Proof. Let (an) be an increasing binary sequence; for each n, let b,=I - an; and let S - (--, 0] U (Un=,[1 + 2-

o)).

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DOUGLAS S. BRIDGES

Note that if a, = 0 for all n, then S = (--,, a, = 1, then S = (--,,

0] U [1, -c); and that if

0] U [1 + 2k, o-) for some k < N. According to

the remarks preceding (7.3), to prove that S is located in R it will suffice to show that for all x in the dense subset (-co,

0] U (0,11) U 41 1) U [1,

c)

of R, and for all real numbers (x, 13with 0 < (x < 13,dist(x, S) exists, or Ix - sl >Ža for all s in S, or there exists s in S such that Ix - sl < 1P.If x E (-o-, 0], then dist(x, S) = 0; if 0 < x < 1/2, then dist(x, S) = x. In the case 1/2 < x < 1, we may assume that 13 < 1. Either x < 1 - x, in which case 0 < ax < 1/2, and Ix - YA> x for all y E S; or x > 1 - 13.In the latter case, choosing a positive integer N so that 2-N < x + 13- 1, we see that if aN = 0, then there exists s E S such that 1 < s < x + 13 and therefore Ix - sl < 13; whereas if aN = 1, then S is located. Finally, if x > 1, then, choosing a positive integer v so that 2-v < x, we see that if av = 0, then dist(x, S) = 0; whereas if av = 1, then S is located. This completes the proof that S is located. Now suppose that T =s n (-1o, 1] is located in IR, and let 8 dist(2, T). Either 8 < 2, in which case, as (0, 1) C R - S, 1 must belong to T, and therefore a. = 0 for all n; or else 8 > 1, in which case 1 0 T and so -,Vn(a, = 0). Thus (i) entails WLPO. By considering {x E R : -x c S), we can prove that (ii) entails WLPO. Finally, to prove that (iii) entails WLPO, we use the fact that if S n [0, 1] is totally bounded, then dist(2, S n (--c, 1]) exists. 0 An alternative, more intricate, characterisation of closed located subsets of R is given in Mandelkern (1981), and depends on the construction of the extended real line (that is, R• with - and -•c adjoined to it, together with an appropriate metric). 13.

CONVEXITY

At first, it might be imagined that there is little of interest to say constructively about convex subsets of lR. However, that topic is an excellent illustration of the phenomenon of branching of concepts, in which a single classical concept gives rise to several interesting constructive ones that are classically, but not constructively, equivalent.

A CONSTRUCTIVE LOOK AT THE REAL NUMBER LINE

77

A subset S of R is said to be convex if tx + (1 - t)y belongs to S whenever x, y e S and 0 < t < 1; weakly convex if tx + (1 - t)y belongs to S whenever x, y E S and 0 < t < 1; paraconvex if [a, b] C S whenever a, b E S and a < b; weakly paraconvex if [a, b] C S whenever a, b E S and a < b; ultraweakly paraconvex if (a, b) C S whenever a, b e S and a < b. These properties of S C R are easily seen to be equivalent classically; constructively, we have the following result. (13.1) Let S be a subset ofR . Then (i) if S is either convex or paraconvex, it is weakly paraconvex; (ii) if S is weakly paraconvex, it is ultraweakly paraconvex; (iii) S is weakly convex if and only if it is ultraweakly paraconvex. Proof. Most of the conclusions are trivial to establish. Those that are not trivial follow readily from the following observation: if a, b are points of S with a < b, then for each x in [a, b] we have x = ta + (1- t)b, where 0 < t a (b - x)/(b - a) < 1. E We now show that the constructive relationships described in the last result are the best we can hope for. (13.2) Each of the following statements entails LLPO: (i) Every convex subset of R is paraconvex. (ii) Every paraconvex subset of R is convex. Each of the following statements entails LPO: (iii) Every weakly convex subset of R is convex. (iv) Every ultraweakly paraconvex (or, equivalently, weakly convex) subset of R is weakly paraconvex." Proof. Let (aj)be a binary sequence with at most one term equal to 1, and define another binary sequence (b.) by bý = a. = 0

if either a, = 0, or else a, = I for some even k < n, otherwise.

78

DOUGLAS S. BRIDGES

Let a aY- =,a,2-' and b = XY-tb,,2- ; so 0 < b < a. Note that if b > 0, then a, = 0 for all odd n; whereas if b < a, then a, = 0 for all even n. The set T-

{ta :0• t < 1}

is clearly convex. Suppose T is paraconvex; then b E T, so b = ta for some t E [0, 1]. Either t >0 or t < 1. In the first case, if a= I for some odd k, then b = ta > 0, a contradiction; thus a. = 0 for all odd n. In the case t < 1, if a, = I for some even k, then b = ta < a, a contradiction; so a. = 0 for all even n. Thus (i) implies LLPO. Now let x E R•, and consider the paraconvex set S =_

E

:s

_ t for some sand tin {0, x}}.

Clearly, 0 and x belong to S. If S is convex, then there exist s, t in {0, x} such that s < 1/20 + '/2x < t. If s = 0, then x > 0; while if s =x, then x < '/2x and so x _<0. Hence, (ii) entails Vx E R (x < 0 or x > 0), which is equivalent to LLPO. Next, consider the set A = {0} U (0, 1). It is clear that A is ultraweakly paraconvex and therefore weakly convex. If A is either convex or weakly paraconvex, then [0, 1) C A, from which it readily follows that Vx E Rf(x Ž 0 • x = 0 or x > 0). Hence both (iii) and (iv) entail LPO. U Clearly, intervals in R are both convex and paraconvex. Conversely, (13.3) If S is a bounded located subset of R that is either convex or paraconvex, then S- is an interval. Proof. Being located and bounded, S is totally bounded, by (9.5); so, by (9.2), m - inf S and M =-sup S exist. Clearly, S- C [m, M]. On the other hand, if x r [m, M] and 0 < r = dist(x, S), then m + r < x < M - r; choosing s, t e S such that s < m + r and t > M - r, we see that x E (s, t) C S, as S is weakly paraconvex by (13.1); hence dist(x, S) = 0, a contradiction. Thus, in fact, dist(x, S) = 0 for all x in [m, M]; so that [m, M] C 9-. M Note that the closure of a convex (respectively, paraconvex) subset of R is also convex (respectively, paraconvex). The next two results show that we cannot omit either the hypothesis of locatedness or that of boundedness from (13.3).

A CONSTRUCTIVE LOOK AT THE REAL NUMBER LINE

(13.4)

79

The statement every nonvoid bounded closed subset of R that is both convex and paraconvex is an interval

entails LPO. Proof. Let (a.) be an increasing binary sequence, and define Sn = {0) = [0, 1]

if an = 0, if an= I-

Then U_.,S, is nonvoid, bounded, convex, and paraconvex; as is therefore its closure S in R. To prove, for example, that S is paraconvex, let a, b E S, a < x < b, and s > 0. Choose a', b' in U-.=,Sn such that Ia- a'l < E and Ib- b'-< e; then, as S 1C S2 C .... we have a' E SN and b' E S, for some N. If an = 0, then a' = b' = 0, 0 < a < 8, 0 < b <E, and so Ix

-

a'I

Ix - al + a < (b

-

a) + , < 2e;

if aN = 1, then S = [0, 1], and so x e S. In either case, there exists x' E S such that Ix - x'j < 2e. As e is arbitrary and S is closed, we conclude that x ( S. If S is an interval, then cr =-sup S exists. If a > 0, then an = 1 for some n; while if a < 1, then an = 0 for all n. (13.5)

The statement every nonvoid closed located subset of R that is both convex and paraconvex is an interval

entails LPO. Proof. Let (an) be an increasing binary sequence with a. = 0, and for each positive integer n define Sn = [0, n] = [0, k]

if an = 0, if k < n, a,-, = 0, and ak = 1.

Let S =- U.n=1 S.. We show that S, which is clearly nonvoid, is closed and located in R. Let (sn) be a sequence in S converging to a limit sin R, and choose a positive integer N > s_. If a, = 0, then s- E [0, N] C S; if a, = 1, then S = [0, k] for some positive integer k < N, S is closed in R, and so S- E S. Thus S is closed in R. On the other hand, if x < 0, then dist(x, S) = IxI exists. If x > 0, then, choosing a positive

80

DOUGLAS S. BRIDGES

integer v > x, we have either x E [0, v] C S, in which case dist(x, S) = 0; or else.S = [0, k] for some positive integer k < v, and S is located. Thus dist(x, S) exists for all x in the dense subset {x E R : x < 0 or x > 0) of R; whence S is located in R. It is easy to see that S is both convex and paraconvex. If S is an interval, then either S is bounded or S = [0, -o). In the first case, we can find n such that a. = 1; in the second, a, = 0 for all n. U 14.

CONNECTED SETS

We end this paper by discussing, in some detail, constructive properties of the line that are classically equivalent to connectedness. This discussion will bring together various strands from the preceding sections, and will reinforce the point, made earlier, that a single classical concept may split into constructively distinct ones, each of which is of interest on its own. Classically, a subset S of the real line is said to be connected if the only subsets of S that are both open and closed in S are 0 and S. Classical logic alone then suffices to justify the equivalence of each of the following properties of a set S C R: C1 S'is connected. C2 S cannot be expressed as the union of two disjoint nonvoid open subsets. C3 S cannot be expressed as the union of two disjoint nonvoid closed subsets. C4 If A is a proper open subset of S, then there exists a point ý in A- N S such that ý # x for each x in A. C5 If A is a proper closed subset of S, then A n (S-A)- is nonvoid. It then turns out that a subset of the line is connected if and only if it is an interval (Dieudonnd, 1960, 3.19.1). Since, as we shall see, properties Cl-C5 cannot be proved equivalent by constructive logic alone, the constructive relation between these conditions (appropriately modified for constructive analysis) is more complex, and perhaps more intriguing, than the classical. Moreover, there is the possibility that some of the corresponding constructive properties will hold for intervals in R, and some will not. In fact, as we shall show in this section, there are some surprising results on these topics.

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81

We begin our development of a constructive theory of connectedness by introducing properties analogous to C2 and C3: CC2 If S = U U V, where U, V are nonvoid and open in S, then U n V is nonvoid. CC3 If S = U U V, where U, V are nonvoid and closed in S, then U n V is nonvoid. (14.1) If a subset S of R satisfies CC3, then it satisfies CC2. Proof. Suppose S satisfies CC3, and let S = U U V, where U, V are nonvoid open subsets of S. Then S n U- and S n V- are nonvoid and closed in S, and S = (S n U-) U (S n V-). So, by CC3, S n U- n Vcontains a point ý. Either ý E U or ee V. In the first case, choose r > 0 such that y e U whenever y e S and 14 - yJ < r. Then, as 4 V-, there exists y E V such that I- yI < r; whenceY U n v. The case E E V is similar. E Perhaps surprisingly, we cannot expect to prove that CC2 is equivalent to CC3; before justifying this claim, we prove: (14.2) An interval in R satisfies CC2 and CC3. Proof Let I be an interval in R, U and V nonvoid closed subsets of I such that I = U U V, u, E U, and v1 E V. Choose a, b in I such that a < b, u, E [a, b], and v, ( [a, b]; this is possible as I is an interval. Construct sequences (un), (vn) of points of I such that for each n, (i) (ii) (iii)

u E [a, b] n U and v, E [a, b] n V; Iu,,l - uI '--LIv_- uni and Iv.,+1 - vJI
Indeed, having constructed u1 . . . . . u, and v . . . . . . vn with the appropriate properties, let w -2-(u, + vJ)E [a, b] C I. If w E U, set u,+ =-w and vn,+ , v,; otherwise, w E V, and we set u,+, = u, and vn+ 1 w. This completes the inductive construction of the sequences (un) and (v,). By (iii), -

jvn

- unI =

)n

-

U11

for all n. Thus, by (ii), whenever m > n we have

82

DOUGLAS S. BRIDGES U.

- un km= n-'lIUk+ -I UkI lure u12k-1V •X -1, iIV, UkI

_-U-

= EZkT7 -±(1/2)k-I v1 - u11 < Ivi - ul Y-7,. (1/2)k = (1/2)'-lIvl - u11, and, similarly, Iv. - vJ, <

(1/2)n-

1V1

-_

U11.

It follows from all this that (un) and (vn) are Cauchy sequences in R that converge to the same limit ý; by (i), ý belongs to U- n V- n [a, b] = U n v n [a, b]. Hence I satisfies CC3. Reference to (14.1) completes the proof. U The need for the points a and b in the above proof arises because we cannot decide, in general, whether u, < v, or u, > V'. Theorem (14.2) has the following corollary. (14.3) Let I be an interval in R, and U, V disjoint subsets of f such that I = U U V and either or

U, V are both open in I U, V are both closed in L

Then either U = I and V = 0, or else U = 0 and V = I. Proof. For example, in the case where U is nonvoid, as U n V = 0, it follows from (14.2) that V = 0. U We now return to the relation between CC2 and CC3. (14.4) The subset S - {0} U (0, 1] of R satisfies CC2; but if S satisfies CC3, then Markov's principle holds. Proof. Let S = U U V, where U, V are nonvoid open subsets of S. Without loss of generality take the case where 0 E U, and compute r > 0 such that [0, r) n S c U; then r12 E U n (0, 11. Let v E V. Then either v = 0, and therefore 0 E U n V; or else v ( (0, 1]. In the latter case, (0, 1] is the union of the nonvoid subsets U n (0, 1] and V n (0, 1], each of which is open in (0, 1]; so, by (14.2), U n V n (0, 1] is nonvoid. Hence S satisfies CC2. Now let (an) be an increasing binary sequence such that a, = 0 and

A CONSTRUCTIVE LOOK AT THE REAL NUMBER LINE

83

-iVn(an = 0). Define nonvoid subsets U, V of S as follows: U V-

n {{0} U (0, l/n] : a. = 0), U {[l/(n + 1), 1] : an = 01.

It is clear that each set of the form {0} U (0, x], with 0 < x < 1, is closed in S; since U is the intersection of a sequence of such sets, U is closed in S. To prove that V is closed in S, consider a sequence (v,) in V that converges to a limit v- in S. If v- = 0, then we see from the definition of V that a, = 0 for all n, a contradiction. Hence, as v- E S, v- > 0 and we can find a positive integer N > l/v_. If aN = 0, then V D [1/(N + 1), 1], SO V_ E V. If a, = 1, then V = [I/k, 1] for some k < N; so V is closed, and v_ E V. Thus, in either case, v- E V; so V is closed in S. Given x in S, we have either x = 0 r U or x > 0. In the latter case, choosing a positive integer N > l/x, we see that if aN = 0, then x C [1/(N + 1), 1] C V; while if a, = 1, then there exists k 0 and we can find a positive integer v > 1/ý. Since 4 E U, we cannot have a, = 0; so a, = 1. Thus if S satisfies CC3, we can prove Markov's principle. 0 It is an open question whether Markov's principle implies the equivalence of CC2 and CC3 for a subset S of R. We now come to a first constructive expression of the classical fact that a connected subset of R is an interval. (14.5) If I is a subset of R that satisfies CC2, a and b are points of I, and a < b, then I n (a, b) is dense in (a, b). Proof. Let a, b E I and a < x < b. Given a positive number e such thata <x- -<x + e
n (--c, x + C), V=-I n (x-e,

cc).

Then U, V are nonvoid open subsets of I, and I = U U V. So U n v is nonvoid; whence there exists 4 E I such that Ix - 4<. < We cannot improve upon the last result:

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DOUGLAS S. BRIDGES

(14.6) Let I - [-1, 0] U [0, 1]. Then I satisfies CC2; but if(a, b) C I whenever a, b e 1 and a < b, then LLPO holds. Proof. Suppose I = U U V, where U, V are nonvoid open subsets of I. Without loss of generality assume that 0 E U. If V n [-1, 0] is nonvoid, then [-1, 0] is the union of the two nonvoid open subsets U n [-1, 0] and V n [-1, 0]; so, by (14.2), [-1, 0] n U n Vis nonvoid. On the other hand, if V n [0, 1] is nonvoid, then, similarly, [0, 1] n U n V is nonvoid. Hence I satisfies CC2. Now suppose that (a, b) C I whenever a, b E I and a < b. Then (-1, 1) C I; so (-1, 1) = (-1, 0] U [0, 1). It readily follows that Vx E M (x _>0 or x _<0), from which we can prove LLPO. Another notion of connectedness classically equivalent to CC2 is disposed of in the next result, which is found in Mandelkern (1982, Ex. 5). (14.7)

The statement if U, V are disjoint open subsets of a bounded open interval I, then there exists x in I such that x 0 U and x e V

entails WLPO. Proof. Let (a.) be an increasing binary sequence with a, = 0, and define J-- {0} U {n : n = a, + 1 for all k}. Then 0 E J, and 1 E J if and only if a, = 0 for all k. Define nonvoid open subsets U, V of I = (0, 3) as follows: U= U•

1

(0, 1 + a.),

V-

Ujzj(2 -j, 3).

If there exists ý E U n V, then there exist n, j such that j c J and 1 +an>2 -j; soan+j> 1 and thereforean=j= 1;hence 1 e J, so that an = 0, a contradiction. Thus U and V are disjoint. Suppose there exists x e I such that x 0 U and x 0 V. Either x> 1 or x < 2. In the first case, if a. = 0 for all n, then x E V = (1, 3), a contradiction; hence -1 Vn(a, = 0). In the second, if a. = 1 for some n, M then x e U = (0, 2), which is absurd; so an = 0 for all n. We now introduce constructive analogues of Cl, C4, and C5 by the following definitions.

A CONSTRUCTIVE LOOK AT THE REAL NUMBER LINE

85

A subset S of R is said to be connected if A = S whenever A C S is open, closed, and located in S; 0-connected if, whenever A C S is open and located in S, and S - A is nonvoid, there exists a point 4 in A- n S such that ý * x for each x in A; C-connected if, whenever A C S is closed and located in S, and S - A is nonvoid, A n (S - A)- is nonvoid. The locatedness condition (which, of course, is automatically satisfied from a classical viewpoint) appears in these definitions for two reasons: first, it is needed in the proofs of many of the results below; secondly, as the following example shows, the recursive interval [0, 1] would not be connected if we dropped the locatedness condition from the definition of 'connected'. (14.8) There exists a nonvoid convex subset A of the recursive interval I = [0, 1] such that A is both open and closed in I, and I - A is nonvoid. Proof. Assume the Church-Markov-Turing thesis. Let (rn) be a Specker sequence in (0, 1/2), and let A be the closure of U•-1 [0, r.] in R. Then A is closed in I. Given x in A, choose 5 > 0 and a positive integer N such that Ix - r.1 _Ž8 for all n > N. There exist a positive integer v and y E [0, rj] such that Ix - yj < 5; let k - max{N, v). Then rk Ž-r, > y > x - 6; whence rk > x + 8 > x and therefore

(x-5, x+5) n IC [0,

rk]

C A.

Hence A is open in I. It is clear that A is convex, that 0 (1/2, 1] C I-A.

E

A, and that 0

Note that in this example, although A is closed in I, and both A and I - A are open in I, I cannot be the union of A and I - A: otherwise, by (14.2), A n (I - A) would be nonvoid, which is absurd. Classically, the three properties introduced before (14.8) are equivalent to each other (and to each of the conditions C1-C5). The most we can say constructively is the following. (14.9) A C-connected subset of R is 0-connected, and an 0-connected subset of R is connected.

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DOUGLAS S. BRIDGES

Proof. Suppose S C lR is C-connected, let A C S be open and located in S, with S - A nonvoid, and let B be the closure of A in S (so B = A- n S). Then B is closed and, by (12.1), located in S; and S - B = S - A is nonvoid. So there exists a point ý in B n (S - B)-. Given x in A, choose r > 0 such that (x - r, x + r) n S C A. Either x # ý or Ix - ýj < r; in the latter case, as E (S - A)-, there exists y in S - A such that Ix - YI < r, so y E A - which is absurd. Thus x • • for each x in A, and therefore S is 0-connected. Now suppose that S C R is 0-connected, let A C S be open, closed, and located in S, and consider any x in S. If dist(x, A) > 0, then, as S is 0-connected, there exists ý in A- n S = A such that •# y for all y in A; so ý # ý, which is absurd. Thus dist(x, A) = 0, and so x E A- n S = A. As x is arbitrary, it follows that A = S. Hence S is connected. 0 We now prove a succession of lemmas that will enable us to show that the above is the best possible constructive result of its kind. (14.10) Let S be a subset of R such that S n [a, b] is dense in [a, b] whenever a and b are points of S such that a < b. Let A C S be located in S, and b r S - A. Then there exist a E A and ý e A- n (S - A)such that either a 5 4 < b or b < ý < a. Proof. With r =-dist(b, A), use (12.6(ii)) to construct a E A such that dist(a, {b - r, b + r}) < r/2. Then either b - 3r/ 2 < a < b - r or b + r < a < b + 3r/2. Taking, for example, the former case (the latter leads to the second alternative of the conclusion of the lemma), we see that as S n [a, b] is dense in [a, b], there exists x, in S n [b - 3r/4, b - r12). Let 8 =- dist(x1, A) and - x, - 8. Then

8

min{x, - (b - r), (b + r) - xj} > r/4.

On the other hand, 8 < x, - a < r; so b - r/2 5 x, + 8 <x, + r < b + r/2, and therefore I(x, + 8) - y[ > r/2 for all y in A. It follows from (12.6(ii)) that dist(4, A) = 0; so ý belongs to A-. On the other hand, as a, b E S

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and a < 4 < b, there exist points of S n (x, - 8, b) arbitrarily close to •; whence, by (12.6(i)), 4 belongs to (S - A)-. U There now follows a second constructive version of the classical theorem that a connected subset of R is an interval. (14.11) If S is a located connected subset of R, a and b are points of S, and a < b, then S n [a, b] is dense in [a, b]. Proof. Given x in [a, b], suppose that 0 < r =_dist(x, S); so a < x. Then T =S n (--o, x] = s n (-oo, x) is nonvoid, open and closed in S, and (by (12.8)) located in R. Hence, as S is connected, T = S, a contradiction. Thus r = 0 and x E S-. For eachE>0wehavex-a< orb-x<eora<x 0, we have either x > a > x - F or x > a. In the latter case we see from (14.11) that S n [a, x) is dense in [a, x]; thus in either case there exists y E A such that y > x - F. Since E is arbitrary, it follows that x = sup A. Thus, by (12.3), A is located in R. As A is closed in S, and b E S - A, S contains a point belonging to A n (S - A)-; clearly, this point must be x. As x is arbitrary, we see that S is weakly paraconvex. Conversely, suppose that S is weakly paraconvex, so that it satisfies

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the first hypothesis of (14.10). Let A C S be closed and S - A nonvoid, and choose b in S - A. Using (14.10), and ý in A- n (S - A)- such that either a 5 • < b or as S is weakly paraconvex, ý E S. As A is closed in E A n (S - A)-. Hence S is C-connected.

located in S, with construct a in A b < • < a. Then, S, it follows that U

An immediate consequence of this theorem and (14.9) is (14.13) An interval in R is C-connected, and therefore both 0-connected and connected. M For the proof of our characterisation of 0-connectedness note that if A is dense in B, and B is located in R, then A is located in R: in fact, dist(x, A) = dist(x, B) for each x in R. (14.14) A located subset S of R is 0-connected if and only if it is ultraweakly paraconvex (that is, (a, b) C S whenever a, b are points of 12 S with a < b). Proof. Suppose S is 0-connected and therefore (by (14.9)) connected, and let a, b be points of S with a < b. Given x in (a, b), let A = S n (-o, x), which is open in S, and let B =-S n (_oo, x]. We see from (14.11) that S n [a, x) is dense in [a, x]; whence x = sup B, and therefore, by (12.3), B is located in R. Also, A is dense in B; so A is located in R. As b E S - A, the 0-connectedness of S enables us to find a point 4 of A- n S such that 4 # y for each y in A. Clearly, • = x and so x e S. Hence S is ultraweakly paraconvex. Conversely, suppose that S is ultraweakly paraconvex, and let A C S be open and located in S, with S - A nonvoid. Let b E S - A, and, using (14.10) above, construct a in A and 4 in A- n (S - A)- such that either a < 4 < b or b < ! a. As A is open is S, there exists r > 0 such that X E A whenever x S and Ix - al < r. If R - al < r, then S - A intersects A, which is absurd; so 4 # a. Thus either a < 4 < b or b < ý < a; whence 4 E S, by the ultraweak paraconvexity of S. So S is 0-connected. M We have already shown, in (13.2), that the equivalence of ultraweak and weak paraconvexity entails LPO. Referring to our proof of that fact, and to (14.12) and (14.14), we now see that

A CONSTRUCTIVE LOOK AT THE REAL NUMBER LINE

(14.15)

89

The statement every located 0-connected subset of R is C-connected

entails LPO. In fact, the subset S {0} U (0, 1) of R is located and 0-connected; but the C-connectedness of S entails LPO (and is therefore false in RUSS). U It follows from this and (14.9) that (14.16)

The proposition if S is a located connected subset of R, then [a, b] C S whenever a, b E S and a < b

entails LPO.

U

Finally, we show that, constructively, connectedness is essentially a weaker property than 0-connectedness. (14.17) The subset S - [-1, 0] U (0, 1] of R is connected; but the 0-connectedness of S entails LPO. Proof. Let A C S be open, closed, and located in S. We first prove that if A n [-1, 0] is nonvoid, then [-1, 01 C A. To this end, let x0 E A n [-1, 0] and x E [-1, 0], and suppose that 0 < r-- dist(x, A). Then either x 0 < x - r or x, > x + r. Taking, for example, the former case, we see that B

A n [-1, x] = A n [-1, x)

is nonvoid, open, and closed in [-1, 0]. On the other hand, it follows from (9.1) that S is totally bounded; since A is located in S, A is therefore totally bounded and so located in R, by (9.4) and (9.3); whence, by (12.8), B is located in R. Hence, as [-1, 0] is connected, B = [-1, 0]; so x E B, a contradiction. Thus, in fact, r = 0, x E A- n [-1, 0], and so, as A is closed in S, x E A. As x is arbitrary, we concluded that [-1, 0] C A. We can show likewise that if A n (0, 1] is nonvoid, then (0, 1] CA. Now fix 4 in A. Either 4 E [-1, 0] or 4 E (0, 1]. In the first case, [-1, 0] C A; so, as A is open in S, A n (0, 1] is nonvoid; whence (0, 1] C A, and therefore A = S. In the case where ý E (0, 1], (0, 1] C

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A; so, as A is closed in S, 0 e A; whence [-1, 0] C A, and again A = S. Thus S is connected. On the other hand, if S is 0-connected, then, by (14.14), it is ultraweakly paraconvex; so (-1, 1) C S, and therefore Vx E R (x > 0 or x • 0). This entails LPO. 0 There is no obvious constructive connection between the properties CC1 and CC2 on the one hand, and C-connectedness, 0-connectedness, and connectedness on the other. For further results on connectedness see Bridges (1979). University of Waikato, Hamilton, New Zealand NOTES The reader interested in that aspect should consult such works as Beeson (1985), Brouwer (Ed. by van Dalen) (1981), Dummett (1977), Heyting (1971), Shanin (1963), van Dalen (1982), and Weyl (1966). 2 See Kleene-Vesley and the article by Troelstra in Barwise. ' Although, in the constructivist view, mathematics precedes logic, it is interesting, and has proved of immense value in mathematical logic and foundational studies, to present axioms for constructive propositional and predicate calculi; such axioms are found in Chapter 7 of Bridges and Richman (1987). 4 In fact, the recursive interpretation of LPO is false in classicalrecursion theory, as it entails the decidability of the halting problem. See Bridges and Richman (1987), Ch. 3 (1.4). ' It is worth remarking here that, according to intuitionists who accept Brouwer's controversial theory of the creating subject, Markov's principle entails the law of excluded middle: see Bridges and Richman (1987), Ch. 5, Section 4. However, not even all intuitionists accept the creating subject as a valid mathematical entity. 6 Not everyone would agree that the interpretation of universal quantification is quite so straightforward: see pp. 12-19 of Dummett (1977). However, a restricted form of the principle __P = P - namely, Markov's principle is accepted by some constructive mathematicians. Markov's principle is the special case in which P has the form 3n(a, = l) for some binary sequence (a,). ' The symbol = means 'is defined to be' or 'is presented as'. 9 That is, an effective enumeration. "15We should beware of thinking that examples like this one force RUSS to be inconsistent with classical mathematics: from a classical point of view, these examples are properly interpreted as saying something true about recursive properties of the set of recursive real numbers. For example, the function f under discussion is, classically, a

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function defined and recursively pointwise, but not uniformly, continuous on the set of recursive real numbers in [0, 1]; in view of the classical uniform continuity theorem, this function cannot be extended to a pointwise continuous function on the whole classical interval [0, 1]. 11 The first part of (13.2) was first proved in Section 10 of Mandelkern (1983). 12 Or, equivalently, weakly convex: see (13.1).

REFERENCES Barwise, J. (Ed.): 1977, Handbook of Mathematical Logic, North-Holland, Amsterdam. Beeson, M. J.: 1985, Foundations of Constructive Mathematics, Springer-Verlag, Berlin. Bishop, Errett: 1967, Foundations of Constructive Analysis, McGraw-Hill, New York. Bishop, Errett: 1970, 'Mathematics as a numerical language', in A. Kino, J. Myhill and R. Vesley (Eds.), Intuitionism and Proof Theory, North-Holland, Amsterdam. Bishop, Errett and Bridges, Douglas: 1985, ConstructiveAnalysis, Grundlehren der math. Wissenschaften, Bd 279, Springer-Verlag, Berlin. Bridges, D.S.: 1979, 'Connectivity properties of metric spaces', PacificJ. Math., 80(2), 325-331. Bridges, Douglas and Richman, Fred: 1987, Varieties of Constructive Mathematics, London. Math. Soc. Lecture Notes, 97, Cambridge Univ. Press. Brouwer, L. E. J.: 1981, Brouwer's Cambridge Lectures on Intuitionism, Dirk van Dalen (Ed.), Cambridge University Press. Dieudonn6, J.: 1960, Foundations of Modern Analysis, Academic Press. Dummett, Michael: 1977, Elements of Intuitionism, Oxford University Press. Goodman, N. and Myhill, J.: 1978, 'Choice implies excluded middle', Z. Math. Logik Grundlagen Math., 23, 461. Heyting, A.: 1971, Intuitionism, 3rd edn., North-Holland, Amsterdam. Kleene, S. C., and Vesley, R. E.: 1965, The Foundations of Intuitionistic Mathematics, North-Holland, Amsterdam. Ko, Ker-i: 1991, Complexity Theory of Real Functions, Birkhauser, Boston. Kushner, B. A.: 1985, Lectures on Constructive Mathematical Analysis, American Mathematical Society, Providence, R.I. Lifschitz, V.: 1982, 'Constructive assertions in an extension of classical mathematics', J. Symbolic Logic, 47, 359-387. Mandelkern, M.: 1981, 'Located sets on the line', Pacific J. Math., 95, 401-409. Mandelkern, M.: 1982, 'Components of an open set', J. Austral. Math. Soc. (Series A), 33, 249-261. Mandelkern, M.: 1983, 'Constructive Continuity', Memoirs of the Amer. Math. Soc., 277. Rogers, Hartley: 1967, Theory of Recursive Functions and Effective Computability, McGraw-Hill, New York. Shanin, N. A.: 1963, 'On the constructive interpretation of mathematical judgments', Amer. Math. Soc. Translations,Series 2, 23, 109-189. Specker, E.: 1949, 'Nicht konstruktiv beweisbare Sdtze der Analysis', J. Symbolic Logic 14, 145-148. Staples, J.: 1971, 'On constructive fields', Proc. London Math. Soc., 23, 753-768.

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Troelstra, A. S.: 1980, 'Intuitionistic extensions of the reals', Nieuw Archiefvoor Wiskunde (3), XXVIII, 63-113. Troelstra, A. S.: 1982, 'Intuitionistic extensions of the reals II', in D. van Dalen, D. Lascar, and J. Smiley, (Eds.), Logic Colloquium '80, North-Holland, pp. 279-310. Troelstra, A. S., and van Dalen, D.: 1988, 1989, Constructivism in Mathematics, NorthHolland, Amsterdam, (Vol. I) and (Vol. 11). van Dalen, D.: 1982, 'Braucht die konstruktive Mathematik Grundlagen?', Jahrber. Deutsch. Math.-Verein, 84, 57-78. Weyl, H.: 1966, Das Kontinuum, Chelsea Publ. Co., New York.

J. H. CONWAY

THE SURREALS AND THE REALS

There have always been problems about our comprehension of the collection of all real numbers. Things became clearer by 1900 when the logicists had succeeded in the project of defining the reals in purely set-theoretical terms. Let us briefly summarise the approach that has now become traditional. The natural numbers 0, 1, 2, 3 .... which had originally been taken either to be well-known things or to be abstractly defined by Peano's axioms, may now be defined set-theoretically as the cardinals of certain sets, or as the finite ordinal numbers in von Neumann's sense. We next define all the integers . . . , -2, -1, 0, 1, 2, . . . in some way - for the moment we regard these details as unimportant - and proceed to construct the rationals by a method that mechanically boils down to taking the rational p/q to be a certain equivalence class of ordered pairs of integers containing the particular pair (p, q). The reals are finally constructed as Dedekind sections (L, R), such things being partitions of the rationals into two sets L, R in such a way that every member of L precedes every member of R. In fact there are several slightly different paths we can take to get to the reals along these lines, and there will be minor problems about the details that will depend on the path one chooses. One of the purposes of this paper is to explore precisely these details. Another is to discuss the advantages and disadvantages of defining the real numbers in a more novel way, namely as particular surreal numbers.

SOME TRADITIONAL PATHS TO THE REALS

I shall suppose that we already have properties, since my main aim in this to simplify the next few stages. To get from the natural numbers to ities just formally adjoin a sign. This I oneself injecting the sign rules "minus

the natural numbers and all their section is to point out some ways the (signed) integers most authorfeel is both complicated (one finds times minus = plus" and so getting

93 P. Ehrlich (ed.), Real Numbers, Generalizationsof the Reals, and Theories of Continua, 93-103. © 1994 Kluwer Academic Publishers. Printedin the Netherlands.

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embroiled in cases) and morally wrong (surely one should DISCOVER those sign rules). Here is a neat trick. Let us define a translation to be a function defined for some of the natural numbers that satisfies f(x + n) = f(x) + n for all natural number n. [This is to be interpreted as saying that whenever the right hand side is defined, so is the left, and they are equal.] Then it is easy to see that any two translations have a common point of definition (for if f is defined at a, and g at b, then they are both defined at a + b), and that if they agree anywhere, then they agree everywhere that they are both defined. This enables us to say that there is a unique maximally defined translation t! that agrees at any point with a given translation t. We call these integers, and define addition of integers to be composition followed by maximization. For example +k is the translation defined for all n, and taking n to n + k, -k is the inverse to this, which is defined only for numbers _- k. So if we follow -4 by +3, we get the translation that takes 4to3, 5 to 4, 6to5, and so on, and the maximally defined translation extending this is -1, as defined above. The associative law is trivial since we are just composing maps, and the other arithmetical laws reduce to the similar laws for natural numbers. For example, if f(a) and g(b) and both defined, then g(f(a + b)) = g(f(a) + b) = f(a) + g(b) ..... f(g(a + b)), which proves that g + f =f+ g. In this brief description, I ignore multiplication, remarking only that it gives no real problems. We can pass from the integers to the rationals by essentially the same trick. A scaling function (scaler) is a function defined for some integers not all zero (and taking integer values) such that f(nx + my) = nf(x) + mf(y), (again in the sense that whenever the RHS is defined, so is the LHS).

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Again any two scalers f and g have a common non-zero point of definition (for if f(a) and g(b) are both defined, so are f(ab) and g(ab)), and there is a unique maximally defined scalarf! that agrees with a given scalar f at at least one non-zero point. We can use the notation plq for the maximal scalar that is defined at the non-zero integer q and takes the value p there. We define f + g to be the maximization of the function whose value at x is f(x) + g(x) for every x where f and g are both defined, and fg by maximizing the composition off and g, and again the formal laws are easy. I prove that the sum of f = p/q and g = ris is (ps + qr)/qs: We have f(q) = p, so f(qs) = ps, and similarly g(qs) = qr. So the sum function takes qs to ps + qr, and must be the unique maximal scalar that does so. About the step from rationals to reals I have only a few things to say. The first is that it seems simplest in practice to define a real number to be any ordered pair of non-empty rational sets L, R such that: R is the set of all numbers that are greater than or equal to all members of L, and similarly L is the set of all numbers that are less than or equal to all members of R.

This is unorthodox, since it allows L and R to intersect, though in at most one point. The definitions of the arithmetical operations are much simpler, however. My final comment is that there is a really big problem with signs here, that actually makes it quite hard to define multiplication. Most authors split the argument into cases, which I think is morally wrong - why did we take the trouble to adjoin signs properly in the passage from natural numbers to integers if we are only going to mess it all up now? One solution to this problem, which I used sometimes when teaching in Cambridge, is to take this last question seriously, DON'T introduce the signed integers! Instead, proceed from the natural numbers to the non-negative rationals (or the strictly positive ones if you prefer), then construct the non-negative (or positive) reals from these, so having no sign-problem, and then construct signed reals from these in the way that we constructed the signed integers from the natural numbers. I think that this is in fact the simplest way to construct the real numbers along traditional lines. I remark however that there is an alternative solution to this sign-problem: the 'surreal' definition of multiplication works for arbitrary signs. This leads me to my main topic.

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SURREAL NUMBERS

The 'surreals' are a very large class of numbers that includes the reals as its earliest constructed members. There are lots of other surreal numbers, including both infinite and infinitesimal ones. In (Conway, 1976), henceforth ONAG, I defined the surreal numbers (there called simple 'numbers') in the following ways: If L and R are any two sets of numbers, and we do not have x > y and for any x in L, y in R, then there is a number {LIR}. All numbers are constructed in this way. By a convenient although initially confusing convention, if z = {LIR} is a number defined in this way, we write ZL for the typical member of L and zR for the typical member of R, and then merely indicate sets by their typical members, so that z = {ZLIZR}. The order relations and arithmetical operations on numbers are now easily defined in this notation. x = y holds just if we have no yL : x and y ----no x =y just if x -y and y -- x; x > y just if x -y but not y -- x; x = x;

xR;

x + y = {XL + y, x + yL1xR + y, x + yR}; -x = {-XRI_ XL-.

and finally, the product xy is defined to be f xLy + xyL - XLyL, xRy + xyR IxLy + xyR -xLyR, xRy + xyL

-

XRYL.

DISCUSSION OF THE SURREALS

The first thing to be said is that this is a remarkably small set of definitions. It is also true that the proofs of the properties of order and the formal laws of arithmetic are all very short, most of them being of the 'one-line' form, for example y + x = {yL + x, y + xl1 . =x

..

= {x + yL, xL + yl ...I

+ y.

Here the leftmost equality is just the definition of y + x, the middle one is inductive, and the rightmost one is from the definition of x + y. Let us analyse the induction a bit more closely. The number of y can be given by a definition y = {yLIyRI in terms

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of some simpler numbers y', y. So when we come to prove the equality y + x = x + y, it is legitimate to suppose that we have already proved the simpler equalities y' + x = x + y' and yR + x = x + yR, and, for a similar reason, also y + XL = XL + y and y + xR = x' + y. But the expression obtained by making these substitutions in our definition of y + x is just the definition of x + y (with the typical elements of the appropriate L and R written in a different order, which does not matter, since L and R are just sets). THE LOGIC OF THE SURREAL NUMBERS

Many people have some difficulties following these arguments. I shall now discuss some of these. I think it is important to point out that the difficulties are mostly psychological rather than logical, and to confess that many of them are consequences of the style in which I (deliberately) wrote ONAG. What does it mean to say that "all numbers are constructed in this way"? Let me first answer this by giving an analogy. Suppose we were to say that the natural numbers (for me the natural numbers start at 0) are just those obtained by the following two rules: (a) 0 is a natural number (b) If n is a natural number, so is n + 1. Then it seems clear that the only natural numbers are 0, 0 + 1, (0 + 1) + 1, ((0 + 1) + 1) + 1 .... and so on (whatever that means). What it DOES mean was made quite precise by Dedekind and Peano. It means, in fact, that if P is any predicate about natural numbers for which (A) P(0) holds (B) whenever P(n) holds, so does P(n + 1), then P(n) holds for every natural number n. In a similar way, we can now say that the phrase "every (surreal) number is constructed in this way" (that is, by a definition x = {XLIxI}) simply means that if P(x) is any predicate about these numbers for which (c) whenever all the P(xL) and P(xR) hold, then so does P(x) then P(x) holds for every surreal number x. It is now merely a technical matter to verify that this inductive

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principle entails the similar principles in more variables - for example if Q(x, y) is any 2-variable number predicate with the property that (d) whenever all instances of Q(xL, y), Q(xR, y), Q(x, yL), Q(x, yR) hold, then Q(x, y) holds, then Q(x, y) holds for all pairs x, y of surreal numbers. Another point that is confusing to beginners is that the inductions have no 'base'. In fact the earliest number to be constructed must be that given by taking L and R to be the empty set (since there can be no previously constructed numbers). This DOES define a number (since indeed no member of L is -- any member of R), and it is this number that we call 0. It is nice to see some of the inductions working through 0. For example the typical members of the L set for x + y are precisely the numbers of the form xL + y and x + yL obtained by adding one of x and y to a member of the left set for the other. But when x and y are both 0 there are no numbers of either form xL or yL, and so the left set of 0 + 0 is empty, as is the right set by a similar argument. So indeed, 0 + 0 = 0. I think the only other tricky point is that, in this system, equality is a defined relation. The working mathematician is so accustomed to thinking that equality means identity that he or she finds it hard to reason in a system where this is not so. Many people have said that they would have found ONAG easier to understand if I had first defined some things called (say) prenumbers - which would be just the 'numbers' of ONAG - and an equivalence relation on them - the equality relation of ONAG - and then defined a number to be an equivalence class of prenumbers. There were several reason why I thought it better not to work in this way. Occam's razor sliced off the alternative method just discussed, which involves more primitive concepts. A more serious reason is that the equivalence classes involved are actually proper classes in most set theories. However, the real reason is just that my way just seemed more elegant. Logicians will probably have less difficulty here than most mathematicians, since they will have already comprehended the relation between the predicate calculus with and without equality. It seems to me obvious that even in mathematics equality is actually never the same as identity - in the equation a = a the two as are different - one is before the equality sign, and one after! So we might as well be honest and admit that equality can be any equivalence relation we like.

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I want to finish this discussion by mentioning some little technical points. I said that If L and R are any two sets of numbers for which .. then there is a number {LIR} But what IS this number? Again, mathematicians accustomed to working inside Zermelo-Fraenkel set theory usually prefer to spell it out in that theory, and so define a number as an ordered pair (L, R) whose two entries L and R are sets of numbers. We recall in this connection that after Kuratowski, this ordered pair can itself be defined as a set, namely {{L}, IL, R) ). My reason for not doing this is simply that I don't think that the number x we are defining really IS this crazy {{L}, f{ L, R}) thing! I don't even think, for that matter, that the sets L and R really enter into its construction either. I prefer to think of it as being directly constructed from the ELEMENTS of L and R. So, for example, I regard the definition x = tO, 1121 as constructing this x directly from the previous numbers 0, 1, and 2. We don't HAVE to first put 0 and 1 together into a set 10, 11 and also form the singleton {2}, and then combine these into an ordered pair of sets. It's much more natural to tweak the principle that allows us to gather elements into sets so that it now allows us to gather them into what we might call 'bisets', which have both left and right membership relations. Of course, one can do something roughly equivalent to this in Zermelo-Fraenkel set theory, but only at the cost of introducing constructions that are unnatural, irrelevant, and complicated. Occam cuts these out too - the natural environment in which to build the surreal numbers is one in which {. . I • • .} is a primitive construct - this plainly has no greater logical complexity than ordinary set theory, so why should we bother to embed it in that theory? I hope that after these remarks it will be seen that the theory of surreal numbers really is very simple indeed, from a purely logical point of view. We start from what is essentially just set theory, given half-adozen inductive definitions and a dozen inductive proofs (mostly one-liners), and we have the field of all surreal numbers. Since this includes the smaller field of all real numbers, we surely have a much simpler construction of these than the traditional one?

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J. H. CONWAY CONSTRUCTING THE REALS AS PARTICULAR SURREALS

Do We? Yes, I think that indeed we do, but the question will bear discussion. First of all, let's look at some surreal numbers other than the ordinary real numbers. The first one is the number (o= {0, 1, 2, 3, .... I}, where the numbers before I are ALL the natural numbers. This IS a number, since there is no member of L that is >=a member of R, and it is easy to see that 0o>0,

(o>1,

co>2 .....

In fact (o is the simplest infinite number, and as our notation suggests, it may be identified with the first of the infinite ordinal numbers of Georg Cantor. In a similar way, we can construct the negative of wo:

-(0 = 11I0, -1, -2, -3,...., and the reciprocal of w:

1M) = {01l, 1/2, 1/3 ....

}=

{01l, 1/2, 1/4, 1/8 ....

},

along with hosts of other exciting new numbers that are discussed in great detail in ONAG. Is there any easy way to pick out the reals from this plethora of peculiarities? Yes there is. We can define a surreal number x to be real just if it satisfies the conditions -N < x < N for some natural number N, and x= {x-l,x- 1/2, x-1/3,.... Ix+ 1, x+ 1/2, x+ 1/3,...}. It is not hard to see that the collection of such numbers is closed under the usual arithmetic operations, and has the other desired properties for the reals. Is there anything wrong with this? Yes there is. It really IS true that this is a much simpler construction for the real field than any other I have come across, at least from a purely logical point of view. So for the logicist's purpose of 'founding' the reals of set theory it seems to be a very good answer indeed. BUT: (1) There is that problem about equality's being a defined relation. This has actually got a bit worse, since those equivalence classes are

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proper classes, and this seems a bit much, when we are merely constructing the reals. We can hope with this by making the above conditions hold hereditarily, but this is a bit messy. (2) I find it disturbing that in the surreals, 1/4 must be constructed before 1/3. To me it seems that these two numbers are much of a muchness, but the theory insists on regarding 1/3 as being just as complicated as irrational numbers like e and 7t. (3) We must not lose sight of the fact that we construct the reals not only for the purposes of pure logic and logicism. Usually we are concerned with pedagogy, too. The proposed method has many defects on this score - those subtle inductions will confound many a student. Of course, many another student will be intensely excited by them, and by the great new world of infinite and infinitesimal numbers that comes with them! (4) I feel we should also ask: is that what the reals really are? I shall not stay for an answer. CAN THE SURREALS HELP US WITH THE REALS?

Yes, I think they can. I make only some very brief comments here, because it has been a long time since I worked through these ideas in any detail. I think the main point is that one can take the proofs that are natural to the surreal theory and carry them through to some extent in a version of the traditional real theory. Let's see how this would work. The definition of addition mentions xL + y, and here x' will always be a rational number, but y might be any real one. So we must allow 'Dedekind sections' to mention real numbers in their sets L and R. This DOES make our theory look a little less 'traditional', but there's no real problem, and in fact addition and subtraction can be made to work quite smoothly. The gain comes when we consider multiplication. The surreal definition has the great advantage of working in the same way for all sign-combinations of the numbers involved. This really does enable us to solve the sign-problem that I mentioned earlier. In fact I think that one could write out a version of the traditional theory modified along 'surreal' lines that would turn out to be quite short and simple, if a little unusual. I must admit that I'm not altogether sure it would be worth the effort.

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So let's forget the surreals as regards the pedagogy of the reals! For the purposes of pure logic and logicism they are still fine. Of course their real justification for being is much more important - they are nothing less than the only correct extension of the notion of real number to the infinitely large and the infinitesimally small. POSTSCRIPT 1.

THE 'SEQUENCE'

DEFINITIONS

There are two fairly popular constructions of our two systems in terms of sequences. I feel I should explain why they are not discussed in the text above. In the first construction, the reals are constructed as equivalence classes of Cauchy sequences of rational numbers. The second, which is unrelated, defines the surreals to be well-ordered sequences of signs + and -. Pedagogically, or for the mathematician who just wants to understand the abstract structure of the constructed system, these are fine. But in my view, they both fail from the logistic viewpoint, in much the same way. A Dedekind section is determined by its set L, which could be coded by a set of integers, as could any particular Cauchy sequence of rationals. However, an equivalence class of Cauchy sequences is at the next higher level, coded by a set of sets of integers, or, as mathematical logicians now routinely say, by a set of reals. How strange it is to define a real number to be something that has the logical complexity of a SET of reals! The sign-sequence construction of the surreals has much the same defect, although it is harder to articulate concisely. It also has a worse one, from my point of view. With the construction as phrased in ONAG, the ordinal numbers could actually be defined as certain surreals. If we want to do this, then plainly we should not make the definition of the surreals depend on some previously defined notion of ordinal number! POSTSCRIPT 2.

A NOTE ON SURREAL PROOFS

It is in fact quite hard to find the complete system of proofs for the arithmetic of the surreals, even though the proofs themselves are quite easy to follow once found. One of the reasons is that various results must be proved in exactly the correct sequence, and that this holds even for many results that one might not think required any proof at all. One sees this even when experimenting with particular simple numbers. It

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seems at times as though the system slaps one's hand when one tries to prove something ahead of time. Suppose for example, that you want to prove that 0 < 1. You realise of course, that this is really two statements, 0 -5 1 and the denial of 1 _-5 0. But perhaps you didn't realise that before proving these, you must have previously proved 0 :-5 0? If you didn't, your attempted proof will soon tell you. Of course this statement 0 < I is just half of the statement 0 = 0, which also requires proof. (It is also the other half!) The problems when your propositions involve variables are even worse. The assertion x = y implies x + y = y + z requires proof, as does (say) x = x. It seems that the latter must be proved simultaneously with the denials of x < x' and xR < x, and so on. The reader who does not think that such problems can be hard is invited to find the mistake in my treatment of multiplication in ONAG, and to correct it. (It arose just because I misremembered the correct order in which things should be proved.) Princeton University, New Jersey, U.S.A. REFERENCE Conway, J. H.: 1976, On Numbers and Games, Academic Press, New York.

PART III

EXTENSIONS AND GENERALIZATIONS OF THE REALS: THE 19TH-CENTURY GEOMETRICAL MOTIVATION

GORDON FISHER

VERONESE'S NON-ARCHIMEDEAN LINEAR CONTINUUM

1. INTRODUCTION In 1907 Hans Hahn of Vienna published an article on non-archimedean systems of quantities [1]. The study of such systems, according to Hahn, goes back to Paul du Bois-Reymond and Otto Stolz. (It actually goes back further - consider horn angles in ancient Greece, for example.) The work of du Bois-Reymond was published between 1870 and 1882 (Hahn cites only two articles, 1875 and 1877). That of Stolz appeared from 1879 to 1896 (Hahn cites articles of 1881, 1883 and 1891). Hahn also observes that Rodolfo Bettazzi handles some questions of this kind in his Teoria delle grandezze of 1890 [2], and that Giuseppe Veronese built a geometry without use of the Archimedean axiom in his "mathematically and philosophically significant" Fondamenti di geometria of 1891, which was translated into German by Adolf Schepp with some changes by Veronese in 1894 [3, 4]. Veronese subsequently answered various objections to his work - Hahn cites articles of 1896, 1897 and 1898. Tullio Levi-Civita, as Hahn says, gave an arithmetical representation of veronese's continuum in 1892/1893 and 1898 [5, 6]. Finally, Hahn cites Arthur Schoenflies' article of 1906 [7]. In 1981 I treated the work of Du Bois-Reymond and Stolz, along with numerous others who had introduced certain systems containing actual infinitesimals and infinites, or something akin to them, up through contributions of Emile Borel, G. H. Hardy and Felix Hausdorff [8]. I said then I hoped to speak later about the work initiated by Veronese and continued by Levi-Civita, Schoenflies and Hahn, which I (and LeviCivita) take to be an approach to non-Archimedean systems distinct from those initiated by du Bois-Reymond and Stolz. This is what I propose to do here. Hahn's brief historical remarks don't do justice to the originality and depth of Veronese's work, and are misleading since they seem to imply that Veronese's system was similar to those of du Bois-Reymond and Stolz. Detlef Laugwitz has observed that Hahn's work on non-Archimedean systems owes much to that of Levi-Civita on the same subject [12]. My observation is that Levi-Civita's work on non107 P. Ehrlich(ed.), Real Numbers, Generalizationsof the Reals, and Theories of Continua, 107-145. © 1994 Kluwer Academic Publishers. Printed in the Netherlands.

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Archimedean systems owes much to that of Veronese on the same subject, a debt which Levi-Civita himself acknowledges [5] (cf. Part 3 below). Veronese owed something to Bettazzi, but went far beyond Bettazzi in many respects, and moreover imbedded his algebraic work in an elaborate system of synthetic geometry. Veronese is relatively unknown to mathematicians today, and my purpose is to describe and commemorate one of his contributions to mathematics, the construction of a non-Archimedean linear continuum. There is a biography of Veronese in Appendix I which gives some idea of his contributions to projective and n-dimensional geometry. This Appendix also contains a discussion of some of the philosophical underpinnings of Veronese's work, and of relations to Hilbert's work on the foundations of geometry. In Appendices 2 and 3 there are translations of some remarks of Veronese on what he called the 'intuitive continuum', and of some of his opinions about the continuum of Dedekind and Cantor. Veronese's style is difficult, to say the least, and to appreciate his work, it will be best to start with a summary of Hahn's article.

2.

HAHN'S ALGEBRAIC NON-ARCHIMEDEAN

SYSTEM

By a non-Archimedean system (of quantities or magnitudes, Grossen), Hahn understands a simply (linearly) ordered system in which there is an addition which satisfies six conditions: closure, compatibility with an equality, associativity, commutativity, existence of unique inverses, and compatibility of addition with the order. In short, as we would now say, a (simply or linearly) ordered group, in which the archimedean axiom is not assumed, i.e. it is not assumed that for any two positive a and b with a < b there is a natural number n such that na > b. The quantities of such a system can be put into equivalence classes in each of which the Archimedean Axiom holds, and which are themselves simply ordered. The order type of this set of classes is called by Hahn the class type of the original non-Archimedean system. In Section 1 of his article, Hahn proves that there are non-Archimedean systems of arbitrarily prescribed class type. It has been known for a long time, says Hahn, that the complex numbers (i.e., hypercomplex numbers) with n units (basis elements), simply ordered, furnish examples of non-Archimedean systems of finite class type. Conversely, Bettazzi showed in his work of 1890 that any non-Archimedean system of

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quantities of finite class type can be represented arithmetically by complex numbers with n units. Hahn shows in his Section 2 that any non-Archimedean system of quantities can be represented by complex numbers whose units form a simply ordered set, in general infinite, with the order type of the system of units being the class type of the non-Archimedean system. Hahn observes that his proof assumes that the quantities of the nonArchimedean system can be well-ordered, although he comments that Zermelo's proof (1904) of the well-ordering theorem had been criticized by Poincar6 (1905). In Section 3 of his article, Hahn distinguishes between complete and incomplete non-Archimedean systems of quantities, a subject which we will return to below. We back up now in Hahn's article to consider it in more detail. In Section 1 of his article Hahn observes that for an ordered group, we can consider four mutually exclusive and exhaustive cases. Suppose c and d are positive quantities in an ordered group. We may have: (I), for every multiple nc of c there is a multiple md of d such that md > nc, and conversely for every multiple m'd of d there is a multiple n'c of c such that n'c > m'd; (II) for every multiple of c there is a multiple of d which is greater than the multiple of c, but not conversely; (III) for every multiple of d there is a multiple of c which is greater than the multiple of d, but not conversely; (IV) it is neither the case that for every multiple of c there is a multiple of d greater than the multiple of c, nor that for every multiple of d there is a multiple of c greater than the multiple of d. Case (IV), however, is never realized, since it implies the existence of c and d such that c > d and d < c. In Case (I), we say c and d have the same height; in Case (II), that c is lower than d; in Case (III), that c is higher than d. One sees that in Case (II), every multiple of c is less than d. In fact, if mc > d, then for every n, nmc > nd, so for every multiple of d there is a multiple of c greater than the multiple of d, contrary to the assumption that it is not the case that for every multiple of d there is a multiple of c greater than the multiple of d. Similarly, in Case (III), every multiple of d is less than c. One sees that if a is lower than b, then a < b, and if c is higher than d, c > d. To extend to negative quantities, suppose a < 0, so -a > 0, and that b > 0. We then regard a to be lower than, equal in height to, or higher than b according as -a is lower than, equal in height to, or higher than b. Similarly when a < 0 and b < 0, using -a and -b.

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Hahn now decomposes the nonzero elements of his system into disjoint classes, such that in any class the elements have the same height. The zero element is adjoined to each of these classes. We can then define a simple order on the set of classes by setting A < B if and only if any quantity in A is lower than any quantity in B. The set of classes has a Cantorian order type, and this is the class type of the non-Archimedean system. An Archimedean system is a special kind of non-Archimedean system, of class type 1. The complex numbers are a non-Archimedean system of class type 2, if we set a + bi > a' + b'i when a > a', and a + bi > a + b'i when b > b'. Then every pure imaginary is lower than every other complex number. This construction can be quite easily extended to (hyper)complex numbers having n basis elements, using coordinatewise addition (this was the result of Bettazzi, 1890), and Hahn proceeds to extend the process to construct non-Archimedean systems of arbitrary class type. In Section 2 of his article, Hahn proves that every non-Archimedean ordered Abelian group can be realized as such a system of complex numbers, of some class type. The proof is lengthy, and uses the wellordering property. Hahn states the final result as follows: "The quantities of an arbitrary non-Archimedean system of quantities [i.e., simply ordered group] can be expressed as complex numbers, whose units form an ordered set F, and whose order type is the class type of the nonArchimedean system. In each of these complex numbers, the units with nonzero coefficients form in F an increasing well-ordered set. The addition is obtained by adding coefficients of equal units. Of two of these complex numbers, the larger is that in which the first unit that does not have equal coefficients in both complex numbers has the larger coefficient [i.e., the lexicographic ordering]." A salient part of the construction is that to each element of G there is assigned a 'sum' obtained by: (1) taking a set of 'units' ea which form a set F of the same order type as G and which are ordered by 'height'; (2) taking any infinite set of the e, with this order, and assigning a real number to each e,; (3) writing the result as a sum aaleal + . . • + ae•. However, each such sum may be regarded as an infinite sum which contain as summands the other 'units' with coefficient zero, in their order according to height. We return now to the question of completeness, as discussed in Section 3 of Hahn's article. In the process just described for assigning symbols

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to a non-Archimedean system G, it may not happen that there is an element of G for every choice of real numbers in a sum. Then the system is incomplete. When there is an element of G for every such 'sum', the system is complete. Hahn proves that this definition is independent of the choice of well-ordering of the 'units'. We can pick out Hahn's Archimedean systems from the simply ordered groups, i.e. the systems satisfying the six conditions given by Hahn (stated above), by requiring that they be the systems with class type 1. Then, he says, we can pick out from these the complete Archimedean systems by imposing Hilbert's completeness axiom, which he states in the following form: "It shall not be possible by adjoining new quantities to the quantities of our system to obtain a more comprehensive simply ordered system in which the six conditions concerning addition are again possible, without new classes of quantities arising therefrom." The demand is that no extension of an Archimedean system can be made which preserves the axioms for an ordered group. Hilbert's completeness axiom has a considerable history of its own. Hahn refers to the 2nd edition of Hilbert's Grundlagen der Geometrie. In the first edition of 1898, there was a continuity axiom, which was in fact the Archimedean axiom [10]. In the 7th edition of 1930 - the last during Hilbert's lifetime - the completeness axiom, in the section called 'Axioms of Continuity', has become the completeness theorem: "The elements (that is, the points, lines and planes) of the geometry form a system for which it is not possible to make an extension with points, lines and planes which preserves the incidence and order axioms, the first congruence axiom, and the Archimedean axiom; they therefore form all the more a system for which no such extension is possible which preserves all the axioms." There is a footnote indicating that in previous editions this was an axiom, but that Paul Bernays had pointed out that an axiom of linear completeness was sufficient: "The points of a line form a system for which no extension is possible which preserves the linear order. . . ,the first congruence axiom and the Archimedean axiom; that is, it is not possible to adjoin points to this system of points in such a way that in the system arising through this combination all of the axioms introduced are satisfied" [11]. In the English translation of the 10th edition by Leo Unger there is a footnote giving credit to F. Bachmann for an analysis of requirements in the axiom for linear completeness as incorporated in the 7th edition.

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Hahn observes that a complete Archimedean system differs in no essential way from the real number system. That is, any such system is isomorphic to the real number system. As for the complete non-Archimedean systems, Hahn says we need merely require that our system have a class type other than 1, and require the same completeness condition. There remains the question of multiplication and division. In Section 4 of his article, Hahn shows that these can be introduced as follows. Suppose we have a simply ordered group G, and a simply ordered set F of basic elements for elements of G. Assume now that not only G but also F is an ordered group, i.e. that there is an addition on G compatible with its order. As before F and G are to be of the same order type. Given elements a,lei and a,.2er, 2 of G, their product is aala,2e,1+,2. We know e,1 + ,2 is in F since F is a group. Hahn proves that when al and a2 run independently through a well ordered set, the set of all resulting ecl+a2 for a well-ordered set, and furthermore that each o:l + a2 can be obtained as a sum of elements of F in only afinite number of ways. Let A = Yacpeaq and B = Zbapep be two elements of G. To find the coefficient of an arbitrary e,, of G in the product AB of A and B, express cx as a sum of the two elements of 1-"in all possible ways. The number of ways will be finite, as we just remarked, so we will have c = cx(l1) + oc(21) = ... = (x(ln) + cx(2n). The coefficient of eQ in AB is to be aQ( 1 b,)ba( 2 1) + . . . + at(ln)ba(2n). Hahn shows that for a pair of elements g, and g 3 of G, there is, provided g, is not the zero of G, an element g2 of G (quotient of g 3 by g0) such that glg 2 = g 3, where g9g2 is given by the multiplication just defined. Indeed, Hahn shows that G, with its given addition and subtraction, becomes an ordered field (with compatibility of multiplication and order) when multiplication and division are defined in this manner. The construction of multiplicative inverses is rather complicated, and I refer readers to Hahn's paper for details. Finally, Hahn takes the additive identity e 0 of F and identifies the real number system with elements aeo where a is real. Then every positive number of his non-Archimedean system whose class is higher than eQ (see third paragraph of Section 3 of this article) is higher than every multiple na of a, and may be said to be 'actually infinitely large'. Similarly, every multiple by n of a number whose class is lower than e, is 'actually infinitely small'.

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VERONESE'S INTUITIVE CONTINUUM

Tullio Levi-Civita says in his first article on the algebra of nonArchimedean systems: "The illustrious Professor Veronese, in his masterful work Fondamenti di Geometria a pia dimensioni, just as he was able to overcome the obstacles which inveterate prejudices opposed to the development of hypergeometry as a pure science, so, being led by these studies to discuss and reform the principles of all of geometry, he also contributed new and fruitful views in this regard" [5]. Levi-Civita goes on to say that he will consider out of this vast material only the question of infinitely small and infinitely large segments, and how to represent them purely analytically, in abstraction from their geometric origin. This was also the viewpoint of Hahn who took off, as it were, from Levi-Civita's work. Rather than compare Veronese's complicated and clumsy attempt to make an algebra for his non-Archimedean geometric entities with those of Levi-Civita and Hahn, I will instead discuss his geometric continuum, from which his algebraic system, as well as those of Levi-Civita and Hahn, arose. All references in this section are to his main work [3, 4]. Veronese's fundamental mathematical object is the form. Roughly speaking, a form is what is now called an ordered set. Here 'ordered' means linearly (simply, totally) ordered. Veronese also says 'magnitude' or 'quantity' (grandezze, Grosse) for 'form'. He says forms are 'like numbers' (Section 38). In his definition of form (Section 38), Veronese makes a distinction between order and position. Veronese's alleged definition of 'position' reads as follows, in its entirety (Section 9, Def. VI): If the things A and B are different, we can say, even if they are identical, that they have a different position (his italics).

This is followed by a remark: Thus strictly speaking several things can be the same [he says 'not different'] even if

they aren't equal to one another with respect to any of their characteristics other than each of them being a thing. If, however, we call them identical, we disregard their different positions, and if we call them different we disregard their common character-

istics. There is a footnote to this: If new things are defined or constructed by means of things already investigated, it is a

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logical error to define their equality if this word is to retain its original and general meaning, explained here, and if the new things are in themselves completely determined by the definition.

This general sense of 'equality' is, for things A and B (Section 8, Def. VI): "The statement: the thing A is equal to the thing B means: the concept of the thing A is the concept of the thing B." It appears from his definition of "different things" (Section 8, Def. 5, based on Section 2) that things are different if they correspond to "different concepts". Concepts A and B are different if "the concept A is not the concept B" (Section 8, Def. II). As I understand him (Section 8), Veronese bases our ability to decide whether concepts are the same or different on a postulated operation of comparison which, when applied to concepts represented by A and B, yields either 'A is B' or 'A is not B'. This operation permits us to assert of a concept A that 'A is A', of concepts A and B that 'if A is B, then B is A', and of concepts A, B and C that 'If A is B and B is C, then A is C'. In connection with 'A is A', he explains: If A and B represent a single concept c, then the concept represented by A is c and the concept represented by B is C. We say: The concept A is the concept B or is the same concept as B.

Things are identical or absolutely equal (Section 9, Def. III) if they are equal with respect to all of their characteristics. A characteristicof a thing is "that by means of which we can compare it with other things" (Section 9, Def. 1), so it is what the operation of comparison operates on. Equality or difference with respect to a characteristic can be determined by means of comparison. Things are relatively equal or equivalent if they are equal with respect to some of their characteristics. To say that things are equal, tout court, is to say that the concept of the thing A is the concept of the thing B. It appears from all this that two things which are different (judged by comparison to be represented by different concepts) may be identical (judged by comparison to have all the same characteristics). Furthermore, things are different in position if either they aren't identical (differ in at least one property), or are identical (have the same characteristics) but are different (are represented by different concepts). Veronese gives two examples of 'difference in position' when he defines 'form' (Section 38). First: After the concept A is present, I repeat the concept A, and then again the concept A. If

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we take into consideration the time which passed during [not between] each repetition, we get a relation of position which is not contained in the concept of simple sequence

and order, since the time which passed during the first repetition can be different from that which passed during the second.

Second example: I first say the vowel a softly and then the vowel e loudly; the level of voice furnishes a relation of position that isn't contained in the concept of the order in which I pronounce the vowels a and e.

Before considering his system of infinitely large and infinitely small quantities, Veronese spends considerable time building a foundation for the integers and rational numbers, consideration of which I omit here. Section 55 consists of some "empirical considerations about the intuitive rectilinear continuum". Let the word 'continuum' here mean 'rectilinear continuum'. We all know the meaning of 'continuum', Veronese says, because we intuit or visualize (intuiamo, anschauen) it. As mathematicians, we want to find an abstract definition of 'continuum' in which intuition and perception no longer play a part. We want to do this in such a way that the definition can be used in deducing, with logical rigor, properties of the intuitive continuum. The abstract definition, however, may be more general than the intuitive continuum. In the abstract definition, we are permitted to make any assumptions which do not contradict experience (nor, presumably, each other). That is, we mustn't contradict our experiences of the intuitive continuum. When we pick out a part of a continuum, we introduce signs or 'points' to mark the ends of the parts into which the continuum is decomposed. Points are considered to have no parts. We need not consider the points as themselves parts of the continuum, but only as auxiliary mental entities which indicate where parts of the continuum are joined. The continuum itself does not consist of these points. We can consider a point of separation as belonging to either of the two parts determined by the point. For example, suppose parts p and q of a continuum are determined by points P, Q and R in such a way that p is determined by P and Q, q is determined by Q and R, no other points of separation are introduced, and there is no part of the continuum between p and q. We can call P and Q the endpoints of p, and designate the part by (PQ) or (QP). Similarly, we can call Q and R the endpoints of q, and designate q by (QR) or (RQ). We can say that Q belongs to both p and q. But to say.a point belongs to p or q is not to

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say that it is a part of the continuum. It is only to say that Q is assigned to p and q in order to pick p and q out of the continuum. Experience teaches us that with a given experimental procedure, and a given empirical continuum, we can reach after a finite number of decompositions a part of the continuum which can no longer be decomposed. That is, an empirical continuum will contain indivisibles. However, we know too that with the same continuum and another experimental procedure, we may find that the former indivisibles are now divisible. Hence we would like to consider a continuum which has no indivisibles. This in itself precludes an abstract continuum composed of points. Veronese's aim is to give an abstract definition of continuum which incorporates these features. Veronese considers sequences of forms (Sections 56-61). In Veronese's terminology, these are 'ordered groups'. They may be infinite, or, as Veronese says, 'unbounded'. A sequence of forms is considered to be a form itself. A basic element or element of a sequence is a given 'first form'. Different instances of a basic element are said to have a different position (see above). Veronese remarks that "instead of saying 'an element, we can also say 'two or more elements' which 'coincide'." We say of two elements which aren't the same, but can be regarded in a certain way as a single element, that they coincide in this way, or coincide relatively. To coincide absolutely is to coincide in all respects, except possibly in position. It appears then we can have many instances of 'the same' triangle, say, which differ 'in position' by virtue of being different instances, and which may or may not 'coincide'. If we consider a form as 'given', then its definition ('determining law') is called an existence law. If, however, it is considered as constructed, then we have a construction or generating law. Thus 'existence laws' are presumably axioms, and 'construction laws' are definitions or theorems. However, Veronese remarks that if the elements of a form have been arrived at using a law of construction, then the elements can then be considered as given, and "the construction law becomes an existence law"; and if elements of a form have been arrived at using a law of existence, and then are constructed "the existence law becomes a generating law". To relate one 'group of forms' to other forms is "to consider the

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given sequence of forms together with the other forms". The result is a relation ('relation form'). Veronese says (Section 59): If from the definition of a form it doesn't follow that all possible elements belong to the form A, then one can image elements that have a different position than the elements of A (i.e., lie outside of A) and are independent of A.

He adds the remark: In the future, when we speak of a form, we will mean, if nothing else is specified, that it doesn't contain all basic elements.

I leave the reader to apply this stipulation to the set of all sets which aren't elements of themselves. A 1-dimensional system is a form given by an arbitrary sequence of elements and its inverse sequence, which may or may not have a first or last element, and whose order from any element on is a given characteristic of the form (Section 62).

It appears, then, that we can think of a 1-dimensional form as a discrete countable set together with a linear order on the set, and also the 'inverse' linear order, defined in such a way that whenever x precedes y in the original order, y precedes x in the inverse order (Section 33). The given order and the inverse order are called the directionsor sense of the form, and either one is the opposite of the other. A direction is determined by a choice of two elements of the form. If one system is contained in another, a direction in one determines a direction in the other, and we can then speak of the systems as having the same direction. The parts of such a form consists of all finite subsequences of consecutive elements. For example, if .... A, B, C, D.... are the elements of a form, then the parts are these elements, and also AB, BC, CD, etc., ABC, BCD, etc. The parts which consist of more than one element are called segments. Elements A and B which bound a segment (i.e., are first or last elements) are called ends or bounds of the segment. Any segment has two possible directions, corresponding to the directions of the system in which it occurs. A segment AB is indivisible if there are no elements between A and B in the given order. Two segments are consecutive in a given direction if the second end of the first segment is the first element of the second segment in the given direction. A 1-dimensional system is closed if one applies 'the law of con-

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struction' for the system, starting with an element A of the system, and gets the element A again after having obtained all other elements of the system. A closed 1-dimensional system can be considered as a segment whose ends coincide with an arbitrary element of the system. It follows that a 1-dimensional system is open if it has a first element and is unbounded. An open 1-dimensional system is simple if no element is repeated. A closed 1-dimensional system is simple if, starting from any element, no element is repeated until all of the elements have been 'constructed'. If nothing is said to the contrary, open 1-dimensional systems are taken to be unbounded, i.e. to have no first or last element. A simple open system is decomposed by any one of its elements into two unbounded parts that have no other element in common, one of which can be taken as having one of the possible directions and the other the opposite direction. A closed 1-dimensional system can be considered as an unbounded system, starting from any one of its elements, by considering repetitions of its elements as new elements. A 1-dimensional system is homogenous in a direction if given any segment in a given direction, and any element A of the system, there are two segments in the same direction one of which has A as first end and the other of which has A as second end (Section 68). It follows that when such a system is open, or closed but considered as open (see 4 above), it is bounded in both of its directions. One can prove that such a system is simple (see 4 above). Also, it is possible to deduce the existence of a one-to-one correspondence of the elements starting from an element Aand those starting from an element A', and going in the same direction in both cases. Hert in Veronese's argument that indivisible segments, when there are any, are all equal. Let (AB) and (XY) be indivisible segments. Then, by the definition of indivisibility given above (in 4), there are no elements between A and B, nor between X and Y. By the homogeneity, there is an element YI such that (XY 1) and (AB) are equal (or congruent). If (AB) and therefore (XY 1) weren't equal to (XY), then (XY,) is either larger or smaller than (XY). If (XY,) is smaller than (XY), then Y, is in the segment (XY) (by Veronese's earlier definitions). This contradicts the indivisibility of (XY). If (XY) is smaller than (XYI), hence smaller than (AB), the indivisibility of (AB) is contradicted. It can be shown that a 1-dimensional system which is homogeneous

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in one of its directions is also homogenous in the opposite direction. If in a homogeneous 1-dimensional system, the 1-dimensional homogeneous system obtained by taking all the elements starting from any element A is 'identical' (i.e., the same except for position) to the one obtained by starting from A and going in the opposite direction, the system is said to be identical in the position of its parts. In terms of the 'language of motion', which Veronese had introduced earlier, he says that any part of a 1-dimensional system which is identical in the position of its parts "can move or traverse such a system and moreover remain identical with itself or invariant". Thus Veronese appears to be trying to capture, without the notion of curvature, the definition of 'straightness' given in Euclid's Elements: "a straight line is a line [i.e., curve] which lies evenly with the points on itself" (Heath's translation). Veronese assumes that there is a form which is a 1-dimensional system identical in the position of its parts "which serves to determine all other [forms]" (Section 71, Hypotheses I and II). This is called the basic form. All basic forms are said to be identical. In the basic form, an operation of addition of two consecutive segments is introduced, along with subtraction of segments, one contained in and having an endpoint in common with the other, and an ordering of segments based on the addition. If a segment is given, and we have starting from some element of the basic form n consecutive segments equal (congruent) to the given segment, then the resulting segment is a multiple of the given segment, and the given segment is a factor of the resulting segment. This leads to fractions of segments, denoted by expressions of the form (m/n)(AB), where (AB) is a segment directed from A to B (Veronese writes his scalars on the right instead of the left). If starting from some element of the basic form we take an unbounded natural sequence (of the type of the positive integers) of consecutive segments all equal to a given segment, we have a scale whose unit is the given segment. The two ends of each of the segments in a scale are called the dividing elements of the scale. The region of a scale is the unbounded segment of the basic form consisting of all consecutive segments in the direction of the scale (not necessarily in the scale). The dividing elements of a scale can be assigned the natural numbers, except zero is assigned to the beginning element. When a segment has an nth part, we can assign the rational number m/n the mth multiple of this

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part, starting from the first end of the first segment, and continue in this way through the other segments with n/n = 1, (n + 1)/n, (n + 2)/n etc. If a segment (CD) is equal to a segment (AE) whose ends are two elements of the region of the scale with unit (AB), and (AB) < (CD), then there is a natural number n such that n(AB) :• (CD) < (n + 1)(AB). In fact, the ends of the segment (AE) equal to (CD) are either dividing elements not in the same segment of the scale with unit (AB), or they lie in two different segments of the scale - otherwise we wouldn't have (AB) < (CD). The end E is either a division element or an element inside a segment of the scale with unit (AB). If the former, then by the way we constructed a scale using multiples, there is n such that n(AB) = (CD). If the later, the two ends of the segment containing E have been assigned natural numbers in such a way that n(AB) < (CD) < (n + 1)(AB). If two scales have the units (AB) and (A'B') and (A'B') < (AB) (i.e., there is a segment (B'C') such that (A'B') + (B'C') = (AB)), and also a natural number n such that n(A'B') > (AB), then the regions of the two scales are "equal with respect to the sequences of the segments of the two scales" (meaning to be stated in a moment). For proof, Veronese says let A and A' be the beginning elements of the two scales. Let (AC) be a segment congruent to (A'B'), starting at A and in the direction of the scale defined by (AB). Then it follows from a theorem on order and the hypotheses that (AC) < (AB) and n(AC) > (AB). Hence, by another theorem on order and the first of the hypotheses, m(AC) < m(AB) for any natural number m. Therefore, since the region of a scale is defined to contain all segments in its direction, any segment of the region of the scale with unit (AC) (starting from A) is a segment of the region of the scale with unit (AB). Using the second of the hypotheses, it follows that mn(AB) > m(AB). Hence every segment of the region of the scale with unit (AB) is a segment of the region of the scale with unit (AC). Thus any segment of the region of one of the scales is a segment of the region of the other, which is what's meant by the regions being "equal with respect to the sequences of the segments of the two scales". As a corollary, we have that the regions of two scales are equal if the unit of one is a multiple of the unit of the other. It is also the case that if we have the region of a scale whose unit is equal to a given segment (AD), and the region of another scale with unit (A'B'), and there is no natural number n such that n(AD) > (A'B)

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if (AD) < (A'B) or n(A'B) < (AD) if (AD) > (A'B), then the region of the scale of (AD) and the region of the scale of (A'B) aren't equal. In fact, we can assume both scales start at A, with units (AD) and (AB). First suppose there is no n such that n(AD) > (AB). If (AC) < (AB) and there is k such that k(AC) 2-_(AB), there is no m such that m(AD) Ž (AC). For, if there were, we would have km(AD) Ž k(AC) 2- (AB), contrary to hypotheses. Thus the region of the scale with unit (AD) is completely contained in the segment (AB), and indeed in any fraction of (AB) of the form (1/j)(AB). For the second case, in which there is no n such that n(AB) > (AD), the same reasoning shows that the region of the scale with unit (AB) is completely contained in the segment (AD), or in any part of it which has a multiple exceeding (AD). If A and B are two elements in the region of a scale, then the region of the scale starting from A in the direction of (AB) is congruent to ("equal with respect to a sequence of consecutive segments to") the region of the scale to the part of the region of the scale starting at B. In fact, each element X of the region of the scale beginning with B and with unit equal to (AB) is an element of the region of the scale beginning with A and with unit (AB). This is because to say X is in the region of a scale with unit (AB) beginning at A is equivalent to saying the segment (AX) belongs to the region of the scale with unit (AB), and this is turn is equivalent to saying there is a natural number n such that n(AB) > (AX). But this is equivalent to saying there is n such that n(BC) > (BX) where (BC) is equal to (AB), and if there were no such n, we wouldn't have X in the region of the scale beginning at B. In the region of a scale with unit (AB), let (B...) denote the segment starting with B. By the previous paragraph, we have (AB) + (B . . .) congruent to (B . . .). We say (AB) is negligible with respect to (B . . .). It follows that any bounded segment of the region of a scale is negligible with respect to the remaining segment of the region, starting from the second end of the bounded segment. We recall that the basic form is an assumed-to-exist 1-dimensional continuum identical in the position of its parts, and therefore, so to speak, a 1-dimensional homogeneous system in both directions. Veronese now assumes that in a given direction of the basic form there exists at least one element of the form which lies outside of the region of any scale with a bounded segment as unit (Section 82, Hypothesis III). So, if we choose any segment of the basic form as unit, and generate the scale determined by it, consisting of the multiples of the unit segment

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by natural numbers, there is always another element of the basic form which is not one of these multiples. Veronese evidently intends that the segment determined by the beginning element of any such scale and the new element outside the scale be in the same direction as the scale. Otherwise we could trivially satisfy this new hypothesis about the basic form (as Veronese calls it) by taking what amounts to a negative multiple of the unit segment. Veronese discusses what he has in mind as follows (Section 82). Suppose we have an ordered "group" N (roughly speaking, an ordered set) which starting from any one of its elements is "unbounded of the first kind" (i.e., N is in fact an ordinary sequence), and which is given by "consecutive equal segments in the order of the 'group"' (i.e., is generated by taking multiples of a segment of N chosen as unit segment, and taking the region so obtained). Then we can consider this structure N as an element, and take another element N' which is 'identical' with N (i.e., the same except for position), but which lies outside of N, and which as no basic element in common with N. We can furthermore consider the forms N and N' to be in the order NN'. Then NN' is a finite sequence consisting of two sequences N and N' in which every element of N precedes every element of N'. We can construct a finite sequence of three copies of N, namely N, N' and N", in such a way that (NN') is 'congruent' to (N'N"). If we have a bounded segment (AA') with A in N and A' in N', then there is one and only one element A" in N" such the bounded segment A'A" corresponds to AA'. We say that the basic form extends beyond the region of any scale in it. When we start with a scale and take an element outside of it, we denote the latter element with - in combination with some letter of the alphabet. One verifies that a (bounded) segment which has one end in the region of a scale and the other end outside this region is greater than any bounded segment defined by two elements of the scale. In fact, let A be an origin (AAJ) be a unit segment for the scale, and let A- be an element outside the scale. The proof is based on the definition of scale, which has been made in such a way that the Archimedean postulate fails for (AA_). That is, there is no natural number n such that n(AA 1 ) > (AA-), for otherwise A, would be an element of the scale generated by (AA 1). Conversely, when a segment (AB) is greater than any bounded segment

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in a scale which has A in it and the unit (AA,), then B can't be an element of the scale. These results lead to the following (Section 82, (c)): Any segment which generates the region of a scale is smaller than any segment with the same direction as that of the scale, and which has one end in the region of a scale, and the other end outside the region of a scale. Veronese says (Section 82, Definition II): In order to distinguish the segments bounded by ends which generate the region of a scale with arbitrary unit (AA 1) from those which don't generate the scale and are larger than them [i.e., larger than the segments of the scale], we call the firstfinite and the second actually infinitely large or infinitely large with respect to the unit [of the scale]. However, if the second is smaller than the first, we call it actually infinitely small or infinitely small with respect to the given unit. For example, the unit (AA1 ) or an arbitrary bounded segment of a given scale is infinitely small with respect to an infinitely large segment (AA.).

Veronese's 'Hypothesis III', which postulates that there is always an element of the basic form outside the region of any scale is called by him "the hypothesis on the existence of bounded infinitely large segments" (Section 82, Definition III). In a footnote to this 'definition', Veronese observes that this hypothesis fulfills all the conditions for a mathematically possible hypothesis, which at bottom rest not on considerations of a philosophical nature about the origin of mathematical ideas, but on the absence of any contradiction.

The order relations one expects holds for segments in Veronese's extended sense (Section 82). For example, a segment is either finite or infinitely small or infinitely large with respect to another segment, and the addition of two segments sharing only an endpoint which are finite (infinitely small, infinitely large) with respect to another segment yields a segment which is still the same. The region of a scale is infinitely large with respect to any segment in it, and it follows from Hypothesis III that the region lies in a bounded segment which is infinitely large with respect to any segment in the region of the scale, but which isn't the region itself. At this point, Veronese sheds some light on his concept of "different only in position" (see above). He says that a scale is generated in the same way as (natural) numbers are, "with the difference that here we also take into account the difference in position of the different segments and their parts" (Section 83, Definition V). This brings us to Hypothesis IV about the basic form: If one starts

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with an arbitrary unit segment (AA,), and chooses an arbitrary element BI-) so that the segment (AB(-)) is finitely large with respect to (AA,), then there is an element X in (ABI-1) such that (AX) and (XB(-)) are infinitely large with respect to (AA,), and there exists an element A(-) such that the segment (AX), for any such X, is finite with respect to (AA(-)). This leads, after a while, to a definition of orders of infinite largeness (Section 86, Definition II). If we start with a unit segment (AA 1), and take any A(-) of the sort given by Hypothesis IV, we say the segment (AA(-)) is an infinitely large segment of the first order with respect to (AA1 ). If we apply Hypothesis IV to such an (AA(-)), and get an infinitely large segment of the first order with respect to (AA(-)), the result is an infinitely large segpnent of the second order with respect to (AA(-)). And so on, inductively, to get infinitely large segments of the nth order with respect to (AA,). If a segment is infinitely large of the nth order with respect to another segment, the latter segment is infinitely small of the nth order with respect to the former segment. Segments which are finite with respect to each other are said to be segments of the same kind. Segments which are infinitely large of the same order with respect to a given segment are of the same kind, and similarly for the infinitely small. Veronese introduces a notation for numbers corresponding to the 'second ends' of segments which are infinitely large with respect to a unit as follows. For those that are in an infinitely large segment of the first order with respect to a unit (AA 1), we start with - as origin (corresponding to the element A'- above), and writing -1q+ n or -, - n for the numbers corresponding to ends of segments obtained by adjoining the unit (AA,) n times in one direction or the other. Furthermore, we construct the scale with unit (AA-) by taking multiples of this unit by natural numbers n, and we can adjoin multiples of (AA,) to these in either direction. In this case, we say the larger unit contains the smaller unit 0 times. When we iterate this process, we get for a system of second order with respect to (AA1) (so of first order with respect to (AA-,)), numbers of the form -2 + n (or -n) where now the n refers to the adjoining of the units (AA-,). And so on inductively, for a system of mth order with respect to (AA1). At this point, Veronese inserts a comparison of his infinitely large

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numbers with those of Cantor (Section 90). Here I only quote this remark: The difference between our infinitely large number and Cantor's 0o [i.e., little omega, first transfinite ordinal], is that we do not recognize a first infinitely large number .... while for Cantor's infinitely large numbers, (o is the first in an absolute sense. This means therefore: If one of our infinitely large numbers is given, for example, 001, then there are numbers -, - n different from -, which lie between the finite numbers and oo1. (Remark IV of Section 90.)

Finally, Veronese adjoins his Hypothesis V (Section 91): Every infinitely large segment which no longer is of finite order is obtained by applying

the principle of Hypothesis IV an infinitely large number of times, the number being already given or produced by the new segments constructed in this way.

Veronese calls this the "hypothesis on the construction of infinitely large segments of the basic form". Veronese seems to be saying that one must stipulate in advance what infinite number of times Hypothesis IV is to be applied, or one should use the infinitely large numbers being constructed by application of Hypothesis IV to stipulate how many times Hypothesis IV is to be applied. This is as far as I will follow Veronese in his construction of a onedimensional non-Archimedean continuum and its algebra. Veronese's introduction of multiplication and division, and his discussion of continuity are extraordinarily complex. To unravel and make simpler his treatment of multiplication and division, and algebric notation, would be, I think, to follow in the footsteps of Levi-Civita, Schoenflies and Hahn. Detlef Laugwitz has considered the work of Levi-Civita in some detail in his article on the occasion of the 100th anniversary of the birth of Levi-Civita [12]. I have considered the work of Hahn in the earlier part of this article. The work of Schoenflies doesn't seem to warrant separate consideration in this context. The work of Levi-Civita and Hahn, assisted by Schoenflies, amounts to constructing algebras for Veronese's intuitive continuum, and it is the latter I have tried to illuminate. I am appending translations of two sections from Veronese's book to further this aim. To evaluate Veronese's master work, the Fondamenti di geometria of 1891 and its translation Grundzige der Geometrie of 1894, one must remember that the construction of a non-Archimedean linear continuum is only a fraction of this treatise. In the table of contents, it is the Introduction! Part I of the book is an extended treatment of two- and three-dimensional Euclidean and non-Euclidean projective geometries,

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and Part II is a treatment of geometry of four- and n-dimensions. In addition, there is an extensive historical appendix, and extensive historical material in a preface. APPENDIX

1. A BIOGRAPHY OF VERONESE

Giuseppe Veronese (1854-1917) was born in Chioggia, Italy, near Venice. His father, Antonio, was a house painter and his mother, Ottavia Duse, was a cousin of the famous actress, Eleanora Duse. A good guess is that he was a descendant of Paolo Veronese (1528-1588), the famous Venetian painter, but I have no explicit confirmation of this. While still a young boy, however, Giuseppe showed an inclination toward design and painting. The lack of qualified teachers in Chioggia and the financial circumstances of his family prevented him from following a career as an artist. Nevertheless, he continued all his life to study art and to paint. In 1875 Veronese began a study of synthetic projective geometry with Wilhelm Fiedler at the Polytechnic in Zurich. On the advice of Luigi Cremona, he was brought to the University of Rome in 1876 as assistant to the professor of projective geometry, Salvatore Dino. Veronese held this position for four years. During this time, he published his first work, on the mystic hexagram of Pascal. In 1880-1 he was at the University of Leipzig where he was especially influenced by Felix Klein. An extensive and original memoir by Veronese on synthetic projective geometry of n dimensions appeared in the Mathematische Annalen in 1882. He was appointed to the chair of analytic geometry at the University of Padova in 1881, a position which he held until his death. After ten years of assiduous labor, Veronese published in 1891 his vast and profound work on the foundations of geometry, Fondamenti di geometria a pii' dimensioni e a pia specie di unit rettilinee, esposti in forma elementare. A translation into German by Adolf Schepp, Grundziige der Geometrie von mehreren Dimensionen und mehreren Arten gradlinigerEinheiten in elementarer Form entwickelt, appeared in 1894. It is difficult to translate the title of Veronese's book into contemporary mathematical English in a neat way. The phrase "geometria a pi6 dimensioni" is perhaps best rendered by "multidimensional geometry", but this makes it hard to handle "a piO specie di unitO rettilinee" symmetrically. This phrase can be translated awkwardly by

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"with more than one kind of rectilinear unit". Compromising a little, we can say: Foundationsof geometry of several dimensions and several kinds of linear unit, presented in elementary form. This translates the German title rather better than the Italian one. Veronese presented in this work not only n-dimensional projective geometry from first principles in a synthetic and unified way, including the development of non-Euclidean geometries, but also non-Archimedean geometries containing infinitely small and infinitely large segments. It is to these segments that the 'rectilinear units' refer. There followed a number of articles clarifying and defending his non-Archimedean geometry, and a book, Elementi di geometria, for elementary teaching, written in collaboration with Paolo Gazzaniga. An attack of influenza in the winter of 1911-2 left Veronese with severe circulatory disturbances, but his mind remained vigorous until his death in 1917. During his last years he made a considerable effort to spread his ideas, which met with considerable opposition from the beginning. Schoenflies mentions in 1897 opposition by Cantor, Killing, Peano and Vivanti. A review of the Fondamenti by Peano in 1892 was especially scathing. Peano begins by quoting the first three sentences of the main text of the Fondamenti: "1. I think. 2. I think [of] one thing or several things. 3. 1 think [of] first one thing, then one thing." Peano's evident purpose is to emphasize Veronese's use of metaphysical ideas inappropriate in a work on geometry. He goes on to complain of Veronese's tortured and ungrammatical style. This complaint is, alas, justified. Peano lists a number of what he takes to be errors and absurdities, and he concludes his review with these words: And so one could continue at length the enumeration of the absurdities which the author has piled up. But these errors, [and] the lack of precision and rigor throughout the book, deprive it of any value.

Cantor also was incensed at the work, and criticized it bitterly. On the other hand, Levi-Civita thought it masterful, and he and Hans hahn vindicated it to some large degree. Schoenflies at first criticized it in a certain respect, but later altered his view, and attributed his misunderstanding of what Veronese had in mind to an erroneous example Veronese had given. Furthermore, as we will see, Hilbert called it profound, and made use of its ideas in his own foundations of geometry. Maybe Veronese put too much into his book on the foundations of geometry. The original Italian version has 630 pages, the German trans-

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lation has 710. For comparison, the first edition of Hilbert's Grundlagen der Geometrie in 1899 ran to 90 pages, and even the 7th edition of 1930, with ten appendices, has only 326 pages. Hilbert is reported to have said: "One can measure the importance of a scientific work by the number of earlier publications rendered superfluous by it." Veronese's book is referred to in one of two citations made by Hilbert in the two-paragraph introduction to the first edition of his (Hilbert's) book on the foundations of geometry. Perhaps it was the least Hilbert could do, since Hilbert did indeed for many mathematicians make Veronese's work seem superfluous, along with the work of many other geometers, including Euclid himself. The footnote in Hilbert's introduction which refers to Veronese appears after the following two sentences: The formulation of the axioms of geometry and the investigation of their relationships is a problem which has been dealt with since Euclid in many excellent treatise of the mathematical literature. The problem referred to amounts to the logical analysis of our spatial intuition.

(Leo Unger has 'perception of space' in his translation, where I have 'spatial intuition'. But the usual work for 'perception' is 'Wahrnehmung', and given Kant's influence on Hilbert, I think we should stick to the usual translation of 'Anschauung', which I take to be 'intuition', as found in translation into English of Kant's works. Indeed, the first sentence in Hilbert's book on the same page as the introduction, is an epigraph from Kant which contains the word 'Anschauungen', which Unger translates 'intuitions'. Perhaps Unger was worried about the wider use of 'intuition' among many English-speaking mathematicians, to whom an 'intuitive' proof isn't necessarily one which relies on internal or external pictures, but may be one which leaves out geometric or algebraic details assumed to be obvious or clear. Cf. the translation of Hilbert and Cohn-Vossen's Anschauliche Geometrie (1932) by P. Nemenyi as Geometry and the Imagination (1952).) The footnote which gives credit to Veronese appears in the original edition of Hilbert's Grundlagen der Geometrie, and in the first translation into English by Townsend. However, curious to say, it is absent from the 7th German edition of 1930, and from Unger's translation of the 10th edition of 1968. It runs as follows: Man vergleiche die zusammenfassenden und erlauternden Berichte von G. Veronese,

'Grundlagen der Geometrie', deutsch von A. Schepp, Leipzig 1894 (Appendix), and F. Klein, 'Zur ersten Verteilung des Lobachefskiy-Preises', Math. Ann. Bd. 60

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Which is to say: Compare the comprehensive and expository summaries by G. Veronese, German by A. Schepp, Leipzig 1894 (Appendix) and F. Klein 'On the first presentation of the

Lobachefsky prize'.

This is a reference to the extensive historical essay given as an appendix in Veronese's book. On pp. 24-26 of the first edition of Hilbert's Grundlagen there is a section in which Hilbert proves that the Archimedean axiom is independent of the other axioms of his geometry. Hilbert calls the Archimedean axiom 'the continuity axiom' because "it makes possible the introduction of the concept of continuity into geometry". To prove the independence, he begins with the algebraic number field obtained by starting with the number 1 and applying a finite number of times the five operations of addition, substraction, multiplication, division, and taking the square root of I + w 2 , where w is any number already generated by one of these five operations. He then takes the field of algebraic functions in one indeterminate t over this field, again with these five operations (where w is now any function generated by the five operations). The order is obtained by defining a > b or a < b for two functions a and b according to whether the function c = a - b of t is always positive or always negative for sufficiently large t. For any positive rational n, the function n - t is always negative for sufficiently large t, so n < t for all n. Thus the Archimedean axiom fails. There is a footnote at the beginning of this construction by Hilbert which says: G. Veronese hat in seinem tiefsinnigen Werke, Grundzuge der Geometrie, deutsch von A. Schepp, Leipzig 1894, ebenfalls den Versuch gemacht, eine Geometrie aufzubauen, die von dem Archimedischen Axiom unabhangig ist;

that is, G. Veronese in his profound work, Grundziige der Geometrie, German by A. Schepp, Leipzig 1894, also tried to construct a geometry which is independent of the Archimedean axiom.

Tried - 'made the attempt'. Still, Hilbert does say Veronese's book is tiefsinnig, here translated 'profound' (the word tiefsinnig in some

contexts means 'melancholy'). This footnote, by the way, does appear in the later editions of Hilbert's book, and in the English translation by Unger. Hilbert doesn't seem to have been entirely fair to Veronese. Veronese

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has a synthetic construction of a non-Archimedean geometric system, coordinatized with a field. Veronese's translator, Schepp, says that as far as he (Schepp) knows, Veronese is the first to have built foundations for geometry strictly synthetically, including Riemannian geometry as well as Lobachevskian geometry, and not assuming to start with the archimedean postulate. Veronese's system is richer than Hilbert's. If all one wants is a non-Archimedean field, even Hilbert's construction can be simplified - see, for example, Exercise 5.39 on p. 46 of Real and Abstract Analysis, by Edwin Hewitt and Karl Stromberg (1965). As we said earlier, the last sentence of the first paragraph of Hilbert's introduction to his Grundlagen der Geometrie speaks of the finding of axioms for geometry as amounting to the logical analysis of human spatial intuition. This is a major concern for Veronese. Veronese's starting point and constant guide throughout his foundations for geometry is spatial intuition, or spatial visualization, and extensions of it which form logically consistent systems. As we remarked earlier, Hilbert begins his Grundlagender Geometrie with a quotation from Kant: So f'ngt denn alle menschliche Erkenntnis mit Anschauungen an, geht von da zu Begriffen und endigt mit Ideen. (Thus all human knowledge beings with intuitions, goes from there to concepts, and ends with ideas).

Constance Reid reports in her biography of Hilbert (who grew up in Kant's city, Konigsberg) that at the age of 23, Hilbert defended at his public promotion exercise the proposition: "That the objections to Kant's theory of the a priori nature of arithmetical judgments are unfounded". 45 years later, Hilbert spoke on the question of "the part which is played in our understanding by Thinking on the one side and Experience on the other" (quoted by Reid). He observed that the line between the a priori and the experienced must be drawn differently by us than it was by Kant. He believed (according to Reid) that a logic of the axiomatic method, a species of abstract deduction, is a kind of a priori knowledge of reality. If this is so, then he seems to have believed that this logic is not only a process with which our minds other statements - perhaps in a way agreeable with experience, perhaps not - but also a process by means of which we attain necessary truths about the external world - necessary because there is no alternative. Such a logic would give synthetic a priori knowledge, to use Kant's terminology - knowledge which arises from interaction with objects external to us, but which can

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only arise in one way, determined by the constitution of human minds or brains, and hence cannot be compared with anything else by us in order to be corrected. Hilbert also denied the existence of problems which cannot be solved by such a logic. The last words of his speech were: "Wir muissen wissen. Wir werden wissen." ("We must know. We will know") This was in 1930, and at almost the same time, G6del was finishing the article in which, using Hilbert's sense of a mathematical proof, he

gave a mathematical proof that it is impossible to give a mathematical proof of the consistency of mathematics. Kant argues in the Critique of Pure Reason (1781 and 1787), to state it crudely, that space is not an object, but the form of objects as they are perceived by us. External objects appear to us to be 'in' space, but space is not a receptacle for objects. Space is how external objects are organized by virtue of our sensing them. Geometry is said to be 'synthetic' yet 'a priori'. I take this to mean that we can get the basic concepts of geometry from looking at (or perhaps otherwise sensing) external objects, but that these concepts are determined by the way most human minds, or brains, operate when they look at external objects. We could very well perceive two objects of the same material in the same medium near the surface of the earth, but different in volume, falling at two different rates, the larger falling faster. That they do not is seen by looking at the objects. This knowledge is synthetic but a posteriori: it is obtained by looking at external objects under appropriate conditions, but the result of such looking might be otherwise than it is. However, when we look at the objects, we have no choice as to whether or no they appear to be in space: if they do not appear to be in space, we are not looking at them. Thus space is a form imposed on external objects by us. By virtue of its unavoidability, so to speak, our spatial intuition can be used to yield truths about external objects which are absolutely certain - for example (says Kant) that space has three dimensions. Use of logic, on the other hand, cannot yield knowledge about external objects, although it can yield knowledge about representationsof external objects in our minds or brains. That space has three dimensions is not something we can hope or need to prove using logic - we know it is so by virtue of the way we perceive, and the knowledge that this is the way we must perceive - our consciousness of the necessity of geometric theorems. Kant's views serve to introduce the kind of questions which concerned many geometers during the latter half of the 19th century. In a work of

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1882, Paul du Bois-Reymond had spoken, in connection with the nature of the linear continuum, of systems put forth or interpreted by 'idealists' and 'empiricists' (cf. [8]). The question, roughly speaking, was: which geometric entities and relations are or should be abstractions of perceptions (or 'intuitions'), and which are or should be generated wholly in the minds of geometers? Veronese comments in the preface to his Foundationsthat the psychologist Wundt remarked that 'nominalists' and 'realists' would have been better terms than 'idealists' and 'empiricists'. In any case, the question is reminiscent of Kant's problematic: if we grant that geometry is at least in part knowledge of how our minds organize perceptions, and that this knowledge is imposed on us by the structure of our minds or brains, how much freedom do we have, without falling into contradiction, to extend this knowledge with additional impositions made by ourselves independently of external experience? And we can also ask, does this get us any closer to Kant's things-in-themselves, things as they are independently of human perceptions? In the same year, 1882, Moritz Pasch published a book (second edition, 1926) based on lectures he had given since 1873, in which he develops geometry from a few empirically derived assertions, which he calls 'basic' statements (Grundsiitze, 1882) or 'kernel' or 'nuclear' statements (Kernsatze, 1926). However, he makes the point that processes of inference applied to these statements must be independent of this empirical basis (Pasch, 1882, p. 98; 1926, p. 90). While it may be useful to keep in mind the empirical basis - the 'meaning' of the basic statements - it is not necessary to keep it in mind while reasoning with these statements. Furthermore, no conceptual additions should be made to geometry which are not based on external experience. Thus Pasch's answer to the question at the end of the last section would have to be that we have no freedom to extend our geometrical knowledge beyond what can be deduced from empirically-based hypotheses - making deductions introduces no new concepts from within us. For example, Pasch rejects a certain axiom proposed by Felix Klein on the grounds that an observation cannot connect or refer to (or 'cover', beziehen) infinitely many things, and also because we cannot assume that a segment has infinitely many points without stretching the meaning of 'point' too far (Pasch, 1882, p. 127; 1926, p. 115). A point is characterized by Pasch, somewhat in the manner of Euclid, as a body "whose subdivision is not compatible with the limits of observation".

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Veronese, on the other hand, is concerned to defend the introduction of geometrical concepts beyond those that can be arrived at by observation. When is a mathematical hypothesis possible? [he asks] In the field of mathematics, a definition or postulate or well-determined hypothesis is possible if its conditions do not contradict one another, nor logical principles or operations, nor earlier hypotheses already established, nor truths which have been derived from them.

Again, he says, . . . an abstract geometric hypothesis is possible if it is not contrary to axioms necessary to the development of geometry which we take from experience, nor to the properties of spatial intuition in the limited region of this corresponding to our external observations.... Our general space is geometrically possible, it has therefore an abstract reality, by which, however, we do not mean that the external world is in itself a complete representation of this space. Thus we do not necessarily, with the hypothesis of the different linear units which is a consequence of our hypothesis about the actual infinitely large and infinitely small or in other words the independence of geometry from the Axiom V of Archimedes, have to believe in the concrete reality of the actual infinitely large and infinitely small.

Issues of this general kind are discussed by Roberto Torretti in his book Philosophy of Geometry from Riemann to Poincare(1978). Torretti doesn't mention du Bois-Reymond, perhaps because Torretti concentrated on these issues in their relation to non-Euclidean geometry, whereas du Bois-Reymond was concerned in the place cited only with an unorthodox characterization of a linear continuum. Veronese, on the other hand, deals not only with non-Euclidean geometry in his Foundations,but also with spaces of dimension greater than 3 (including an infinity of dimensions), and with an unorthodox construction of a linear continuum, different from those of du Bois-Reymond, the now standard one descended from Dedekind and Cantor (and ultimately from the one given by Euclid), and the non-standard ones introduced by Abraham Robinson. In the first paragraph of his introduction to the Foundations,Veronese says that he thinks the lively disputes of his day about geometry of more than three dimensions were caused by the purely analytic methods which were in use, and by the confusion introduced by the substitution of 'abstract' or 'numerical' manifolds for geometric spaces properly so-called. This, he says, persuaded him to write a book showing how the geometry of spaces of more than three dimensions can be developed like the geometry of the plane and ordinary space, in order to defend the purely geometric concept of such spaces and to make

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them easier and more widespread. To do this, he develops the geometry of lines, planes and ordinary space from first principles. His unorthodox coordinatization of lines is only one step in this process, although important enough to be named in the subtitle of the book. As we noted, Veronese begins his Foundations with a complaint about the use of real numbers as the basis of geometry. He speaks of the substitution of "abstract or numerical manifolds of n dimensions for geometric spaces, properly so-called". This we may take as a reference to Riemann's lecture of 1867, Uber die Hypothesen, welche der Geometrie zugrunde liegen, in which Riemann generalized to higher dimensions the approach to surfaces made by Gauss in his Disquisitiones generates circa superficies curvas of 1828. He is also referring to subsequent work by Hermann von Helmholtz, Sophus Lie, etc. In the first sentence of his preface, Veronese quotes Helmholtz to the effect that the concept of spatial forms that don't correspond to ordinary intuition can only be securely developed with the calculation of analytic geometry. This predilection of Veronese for synthetic methods goes back to his earliest work. In an article of 1884, Veronese has a long footnote in which he describes how he showed how to construct spaces of any number of dimensions synthetically, "without any analytic substrate", in his lectures at the University of Padovh. He also describes his "principle of projection and section" which he had advanced in an earlier article of 1882. The idea here is that a configuration in a space of a given dimension may be represented in a space of some other dimension by means of projections and sections (i.e., intersections). For example, a configuration of six points in a plane can be obtained by projecting the vertices of a 'fundamental pyramid' (i.e., a simplex) in fivedimensional space. Veronese's use of this technique goes back to his earliest articles, which were concerned with Pascal's mystic hexagram. Gino Loria, in his history of geometric theories (1897, 1931) says it is from Veronese's article of 1882 that "one can truly date the establishment of the synthetic projective geometry of hyperspaces" (although Loria finds the germs of the subject in work by Cayley (1846) and Clifford (1878)). In his history of projective geometry (1939), Federico Amodeo says that from the time of this article by Veronese, these studies had an "interesting and wonderful development", especially in Italy, and remarks on works by C. Segre, E. Bertini, G. Loria and F. Severi growing out of Veronese's memoir - "to mention", he says, "only those before 1890".

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Later in the preface to the Foundations, Veronese expands on his defense of synthetic geometry. Spatial intuition, he says, is what furnishes

us with the first geometric objects and their unprovable properties, and therefore the most proper method for geometry is one which always treats figures as figures, works directly with the elements of the figures and separates and unites them so that each truth and each step of a proof is accompanied as far as possible by intuition.

A method for studying the foundations of geometry which does not belong to geometry is "at least artificial and indirect". Such a method is the numerical or analytical one. Works on the hypotheses of geometry (such as those by Riemann, Helmholtz and Lie) did not advance knowledge of the foundations of geometry. They rather assumed a knowledge of analysis and a good part of elementary geometry. For example, he says, if we assume that a distance is representable

by a number, e.g. the value of a quadratic and positive differential expression, this doesn't tell us what distance is: distance is not a number

geometrically, any more than a line, plane or space of three or more dimensions is geometrically the equations or analytic formulas which represent them. To assume that space is a manifold of three dimensions

corresponding to the triples of real numbers is, for Veronese, to beg important questions. One ought not to forget about the question considered here, [he says] that the last word about the foundations of analysis ... has not yet been spoken; for the principles of analysis are not all logically necessary and include a number of true geometric axioms if one applies analysis directly to the study of geometry (for example, the principle of continuity in it different analytic forms) which are not always justified by spatial intuition. The development by the synthetic method, [Veronese says] succeeds directly for general space, which has an actual infinitely large number of dimensions, while analytical geometry has yet to treat directly such a space (xxiii, xxii).

Hilbert space had not yet been introduced. And: Pure analytic geometry has not yet succeeded in handling our hypotheses about infinitely large and infinitely small quantities suitably.

Cantor had introduced his infinitely large or 'transfinite' numbers beginning in 1879. But, as Veronese comments, his infinitely large quantities

obey the ordinary rules for addition and multiplication, which Cantor's do not. And Cantor claimed to have proved that the assumption that

infinitely small quantities exist leads to a contradiction (cf. [8], pp.

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116-122). Veronese examines this 'proof' by Cantor in the historical and critical appendix to the Foundations, in Note IV, entitled 'Remarks on some proofs against the actual infinitely large and infinitely small'. As to the system of du Bois-Reymond, extended by Otto Stolz, and treated later without reference to infinitely large and infinitely small quantities by such mathematicians as Emile Borel, G. H. Hardy and N. Bourbaki (see [8]), Veronese would no doubt have regarded these as not grounded in spatial intuition, however useful such a system might be for studying increase and decrease of real functions, convergence of real series, etc. In the Note IV of the appendix to the Foundations, he comments on the systems of du Bois-Reymond, Stolz and Cantor, and also on work by Rodolfo Bettazzi (1891-2) and Giulio Vivanti (1891), especially in connection with their treatment - or lack of treatment of infinitely small quantities. Hilbert, in his Grundlagen der Geometrie and elsewhere, based the consistency of his axioms for geometry on a model which assumes the consistency of the real numbers, which in turn assumed the consistency of the arithmetic of the natural numbers. Godel showed in 1931 that a finitary proof of such consistency is not possible. This, it appears, would not have upset Veronese as far as the foundations of geometry are concerned, for he thought these should be based on spatial intuition. Veronese maintains that his introduction of space of infinitely many dimensions, actual infinitely large and infinitely small quantities, and non-Euclidean geometries rests on synthetic methods and spatial intuition, not on this or that foundation for analysis. APPENDIX 2.

VERONESE ON THE INTUITIVE CONTINUUM

Translation of the section of Veronese's Fondamenti (and the German version, the Grundlagen) entitled 'Empirical considerations concerning the rectilinear continuum of intuition' (Section 55): [Footnote to the title of the section:] {In order to establish the mathematical concepts, we can very well fall back on empirically obtained knowledge without therefore having to make any use of it later in the definitions themselves and in the proof.} What is the continuum? This is a word whose meaning we understand even without any mathematical definition, since we intuit the continuum in is simplest form as the common characteristic of many concrete things,

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such as, for example, to give some of the simplest, the time and the place occupied in the external neighborhood of the object sketched here, or by a plumb line, if one takes no account of its physical properties and its thickness (in the empirical sense). Noting the particulars of this intuitive continuum, we should approach an abstract definition of the continuum in which intuition or perceived representation of it doesn't enter any more as a necessary part, in such a way that, conversely, this definition can serve abstractly, with complete logical rigor, for the deduction of other properties of this intuitive continuum. That one can give this mathematically abstract definition, we will see later. On the other hand, if the definition of the continuum is not merely nominal and we want it instead to correspond to the intuitive one, it must clearly arise from investigating the intuitive one, even if later the abstract definition, conforming to mathematically possible principles, contains his continuum as a special case. The object in Figure 1 is said to be rectilinear. Examining now the continuum (Figure 1.a), we see that we regard it as composed of a sequence of consecutive identical parts a, b, c, d, etc., placed from left to right, and that this holds within certain limits of observation. The parts are separated by the vertical marks drawn on the object, and they are also continuous. Furthermore, letting one's eyes scan the object from left to right, one sees that the parts a, b, c, d and also ab, bc, cd, etc., and abc, bcd, etc., are identical from left to right and that these characteristics also occur from right to left. One sees also that between two consecutive parts a and b and c of the sequence abcd, etc., there is no other part, while the part b lies between the parts a and c. [Evidently what Veronese meant is that there are no parts between two consecutive parts such as a and b, or b and c.] If one abstracts the part b, then the rectilinear object is no longer continuous. Observation assures us that what holds for the parts a and

a

A

b

B

c

C

Fig. 1.

d

D

E

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b also holds for two arbitrary consecutive parts of an arbitrary part of it, or in Qther words there is no other whole with the same property the rectilinear object has, whose parts separate two consecutive parts of this object, or, better, that separate the place occupied by it (Section 23 and Definition II, Section 25). [I take it that Veronese is here asserting that the linear continuum is characterized by the property that any bounded subinterval of it separates it into two parts which are not connected, or, perhaps (to be somewhat anachronistic), that any closed and bounded subinterval in the interior of any closed and bounded subinterval of its separates the latter interval into two disconnected parts.] We see further that we can experimentally (that is, with a bounded natural sequence [a finite set] of decompositions) as well as abstractly (that is, according to any mathematically possible hypothesis or operation which doesn't contradict the results of experience) arrive at a part which is not further decomposible into parts (indivisible), of which the continuum is composed (as an instant is for time). It is then experience itself which moves us to look for the indivisible in such a way that we cannot obtain it experimentally, because it shows us that a part considered indivisible with respect to one observation is not indivisible with respect to other observations carried out with more exact instruments or under other conditions. If we assume that an indivisible part exists, we see that we can also experimentally consider it indeterminate, and therefore smaller than any given part of the rectilinear object. We must further distinguish a given part a of the continuum from other parts of its in order to be able to consider it independently from the latter, and if we abstract it, then we cannot consider the remaining part bcd etc. in the order bcd, which we denote for the moment by cx, as having a part in common with a. If the part a remains in its place in the continuum, then in order to distinguish it from the part cx, we must imagine something, a sign (point), that serves to indicate the position of the uniting of the two parts, though the property already observed above remains unchanged, namely that there is no part of another whole between a and cx. The sign which separates the part a from the part cx, is therefore a product of the function of abstracting in our mind, and is not a part of the rectilinear object. If a part cx' of oc follows a from left to right, we can repeat the same consideration between a and Wx.From this point of view, we must therefore assume that the sign of separation or uniting between a and ox, even if it belongs to the continuum,

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is not a part of it. One can consider it as belonging to both parts if one considers one of them independent of the other. Indicating with A, B, C etc. the signs of separation between the parts a and b, b and c, c and d etc. (Figure 1), then we can indicate the part b with the symbol (BC), the part c with the symbol (CD) etc. All of this is supported by the following considerations. Let us suppose, for example, the part a of the rectilinear object is painted red, the remaining part o: white, and suppose further there is no other color between the white and the red. That which separates the white from the red can be colored neither white nor red, and therefore cannot be a part of the object, since by assumption all of its parts are either white or red. And this sign of separation of uniting can be considered as belonging either to the white or the red, if one considers them independently of one another. If we now abstract from the colors, we can assume that the sign of separation between the parts a and c belongs to the object itself. [The idea appears to be that a 'point' can belong to a continuum, being assigned to it, but cannot be a part of it]. Another example. We cut a very fine thread at the place indicated by X with the blade of an extremely sharp knife, the two parts a and a' separate (Figure 2.a), and we assume that one can put the thread back together without seeing where the cut was (Figure 2.b), in other words, without a particle of the thread being lost. One produces this, appar-

X

a

X' Fig. 2.a.

a

X

Fig. 2.b.

a'

a'

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ently, if one looks at the thread from a certain distance. If one now considers the part a from right to left as the arrow above a in Figure 2.a indicates then what one sees of the cut is surely not part of the thread, just as what one sees from a body is not part of the body itself. It happens analogously if one looks at the part a' from left to right. If the sign of separation X of the parts a and a', which by assumption belongs to the thread itself, were part of the thread, then looking at a from right to left, one would not see all of this part, since that which separates the part a from a' is only that which one sees in the way indicated above when one supposes the thread put back together. [Footnote to the previous paragraph:] {We see then that the idea of a point that is not part of the continuum, is quite different from a pure abstraction which does not receive its justification from experience itself. Certainly we make use of our ability to abstract, but in mathematics especially, it is impossible not to make use of it. And therefore it is at least unnecessary, also regulating oneself by observation, to agree that a point is the least perceptible thing of the empiricist, as Pasch does.) The hypothesis that the point is not part of the rectilinear continuum (and also has no parts in itself) means that all the points that we can imagine in it, however many that may be, do not constitute the continuum when they are joined together, and choosing a part (XX') as small as one wants of the object (Figure 1) (for time, an instant), however indeterminate, which is to say without X and X' being fixed in our thoughts, intuition tells us that this part is always continuous. [Footnote to the first clause in the parentheses in the last sentence:] {Thus we do not need to establish in the theoretical development of geometry that a point has no parts in itself. By 'in itself', we don't mean that which a thing is independently of us, but that which it is in its mental representation.) If we now let our eye run from right to left or the reverse over the rectilinear object, we see that every point occupies a definite position on it, and beginning from a given point we do not meet it again either from right to left or left to right, which means that rectilinear object has no knots. We see further that a part, however small, for example that indicate by a primed X, apparently indivisible (Figure 2.b) is bounded on the right and left by parts of the continuum, and hence by two points. And since a constant part bounded by two points indivisible with respect to one observation can be not so after another observation, we must admit the

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possibility experimentally that any part bounded by two points which remains always the same in our considerations, still may be divisible into parts. Moreover, if we consider the rectilinear object from A toward the right, we can agree that the series of parts abcd etc., in this order, is unbounded (Def. II, Section 32), because from repeated experiences we must thing that, if not the rectilinear object, then the place it occupies in the external surrounding is part of an unbounded whole. The same from right to left. Furthermore, between two points X and X' indeterminate in position but which don't coincide (Def. V, Section 8), there is always a continuous part. And because the continuum is determined mathematically by its points, we must assume that between any two points, also indeterminate, however close they are, there is always at least one other point distinct from the endpoints (Def. V, Section 8). We are also led to this assumption by the observation that given a point X on the rectilinear object, one can imagine a part (AB) of the latter which contains X and such that A and B get nearer and nearer to X without however coinciding with X, and that therefore we can imagine a part with the endpoints indeterminate as small as one supposes which contains another point X besides the endpoints. Finally, we receive the impression that in the rectilinear object (Figure 1), around one of its points B there are two parts BA and BC such that, considering the first from B toward A and the second from B toward C, they are identical, and that the part (AB) traversed from B to A is identical to the same part traversed from A to B (Def. III, Section 9). Can all these characteristics of the rectilinear object be established abstractly without recourse to intuition? And if so, are they sufficient to distinguish the continuum as an abstract form from other possible forms? Or aren't some of these necessary consequences of others, even though they are obvious? These are the questions we have to answer in this introduction, and we will see that the characteristics indicated above are more than sufficient. APPENDIX 3.

VERONESE ON CANTOR AND DEDEKIND

Translation of Veronese's note on Cantor,Dedekind and aspects of the continuum, occuring in Section 55 of the Fondamenti and Grundlagen, which is about the 'empirical' continuum.

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G. Cantor and Dedekind for example assert in their valuable works that is one starts with a point of the line, the one-to-one relation between the points of this line and the real numbers forming the numerical continuum is arbitrary. They certainly obtain this continuum by means of a sequence of abstract definitions of symbols which, although possible, are arbitrary. It appears also to Dedekind ... that if three non-collinear points A, B, C are given such that the ratios of their distances are algebraic numbers, then the ratios of the points of space to the distance AB can only be algebraic numbers - so that three-dimensional space and therefore the line would be discontinuous. According to Dedekind, the numerical continuum is necessary in order to clarify the idea of the continuum of space. According to us, however, it is the intuitive rectilinear continuum which, by means of the idea of a point without parts, that serves to give us abstract definitions with respect to the continuum itself, of which the numerical continuum is only a special case. In this way, the definitions appear not as a force which keeps our mind in check, but finds its complete justification in the perceptual representation of the continuum. One must take some account of this representation in the discussion of basic concepts, but without leaving the field of pure mathematics (see Preface). Moreover, it would be truly marvelous if an abstract form as complicated as the numerical continuum obtained not only without being guided by the intuitive, but, as is done nowadays by some authors, from mere definitions of symbols, should then find itself in agreement with a representation as simple and primitive as that of the rectilinear continuum. The rectilinear continuum is independent of a system of points which we can imagine there. A system of points, if we think of a point as a sign of separation of two consecutive parts of the line or as the end of one of these parts, can never give the whole intuitive continuum, because a point has no parts. We find only that a system of points can represent the continuum sufficiently in geometrical investigations. The rectilinear continuum is never composed of its points but of segments, which the points join two by two, and which themselves are still continuous. In this way the mystery of continuity is pushed back from a given and constant part of the line to an indeterminate part as small as one likes, which is still always continuous, into which we are not permitted to enter further with our representation. And at bottom the fundamental concept of limit is wrapped up in this mystery. But it is well to mention that mathematically this mystery has no influence,

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because for us a determination of the continuum by means of a welldefined ordered system of points is sufficient. On the other hand, one should observe that a determination by points is incidental, because we have the intuition of the continuum just as well without it. If in fact one considers a point to be without parts, then, as we have said, even if we make the point of a line starting from an origin correspond to all the well-known real numbers, we don't get the whole continuum. If one considers the point as a part as small as one likes but constant, then not even all the rational numbers are representable on the rectilinear segment starting from one of its points as origin. And this segment still remains continuous in our intuition. Finally, if one considers a point as an indefinitely small part, that is, as being in a state of indefinite smallness, then a point corresponds to any real number without any special axiom. Spatial intuition says here at bottom that if (A) is the abstract form corresponding to the place occupied by a rectilinear object, there is no other abstract form (B) of the same nature as (A) one part of which separates two consecutive parts of (A) (Sections 22, 24). To say that a line can be discontinuous and be given by all the points, considered as without parts, that, for example, represent all the algebraic numbers beginning from a given origin, is to assume a fact that is in itself repugnant to the intuition, namely, that the abstract form corresponding to a line can belong to another possible abstract form that contains all the real numbers whose elements (which are parts of it, a, Section 27) separate the elements of the first form. According to this principle, we are not only constrained to assume that starting from a point of the line all other points represent real numbers, but also to assume that there are points in it which correspond eventually to other possible numbers lying between the real numbers, while the other characteristic properties of the line remain intact. I observe also that we consider the indefinitely small part independently of the distinction between rational and irrational numbers, and that the hypothesis that all these parts do not contain at least one number besides (the endpoints is for us too arbitrary and uncertain. Furthermore, if one has a projectile which goes from a point A toward a point B in a straight line, and one divides its path into the sequence of parts 1/2, 1/2 + 1/22. 1/2 + 1/23, .

. .

and if we accompany it in the sequence of these parts, in our thoughts we don't ever see the tip of the projectile leave this sequence. And yet

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we have an idea of the fact that the projectile hits the point B, which is the limit which the tip of the projectile reaches, and (AB) is the limit of the above sequence, in the sense that while the tip X of the projectile remains in the sequence, one approaches indefinitely the point B or that XB becomes as small as one wants. Thus if one has a sequence of consecutive always increasing parts on this line which, starting from a point A do not pass a point B in our field of observation, in order to represent all of this sequence, we would have to leave the representation of the sequence and represent it as limited by another point C lying between A and B but outside the sequence, if (AB) itself is not the limit of the sequence. And also in this case, the contrary hypothesis would be repugnant to the intuition. One notes, moreover, that the intuition is without doubt essential for geometry, even though it ought not enter as a necessary component either in the statements of properties or of definitions, or in proofs. It has not yet been demonstrated, as far as we know, that there are discontinuous systems of points which satisfy all the properties of space given by experience. In any case, this would say nothing against the continuity of space. For the reasons one doesn't have to place the numerical continuum at the basis of the foundations of geometry, one may see the Preface and Appendix and Section 123 of this Introduction (see also note, p. 86 and note, p. 97). Mathematics and Computer Science, James Madison University, Harrisonburg, VA 22807, U.S.A. REFERENCES 1. Hahn, Hans: 1907, 'Ober die nichtarchimedischen Griossensysteme', Sitzungsberichte der mathematisch-naturwissenschaftlichen Klasse der Wissenschaften, Wien, Abteilung 2a, 116, 601-655. 2. Bettazzi, Rodolfo: 1890, Teoria delle grandezze, Pisa (Hahn says 1891, but 1890 is correct.) 3. Veronese, Giuseppe: 1891, Fondamenti di geometria, Padova. 4. Veronese, Giuseppe: 1894, Grundziage der Geometrie, translated by Adolf Schepp, Leipzig. 5. Levi-Civita, Tullio: 1892/1893, 'Sugli infiniti ed infinitesimi attuali quali elementi analitici', Atti del R. Istituto Veneto di Scienze (vol. 51), series 7, 4(2), 1765-1815.

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8. 9.

10. 11. 12.

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Levi-Civita, Tullio: 1898, 'Sui numeri transfiniti', Accademia Nazionale dei Lincei, Rendiconti, 7 (Series 5A), 91-96 and 113-121. Schoenflies, Arthur: 1906, 'Ober die M6glichkeit einer projektiven Geometrie bei transfiniter (nicht archimedischer) Massbestimmung', Jarhresberichtder Deutsche Mathematikier Vereinigung, 15, 26-41. Fisher, Gordon: 1981, 'The Infinite and Infinitesimal Quantities of du Bois-Reymond and their Reception', Archive for History of Exact Sciences, 24(2), 101-164. Veronese, Giuseppe: 1889, 'I1 contniuo rettilineo e l'assioma V d'Archimede', Atti, Accademia nazionale dei Lincei, Roma, Classe di scienze naturale,fisiche, matematiche, 6 (Series 4), 603-624. Hilbert, David: 1899, 'Grundlagen der Geometrie', in Festschrift zur Feier der Enthullung des Gauss-Weber-Denkmals in Gottingen, p. 19. Hilbert, David: 1930, Grundlagen der Geometrie, pp. 30-31. Laugwitz, Detlef: 1975, 'Tullio Levi-Civita's work on Nonarchimedean Structures', Accademia Nazionale dei Lincei, Atti dei Convegni Lincei, 8, 297-312.

HENRI POINCARt

REVIEW OF HILBERT'S FOUNDATIONS

OF GEOMETRY*

What are the fundamental principles of geometry? what is its origin? its nature? its scope? These are questions which have at all times engaged the attention of mathematicians and thinkers, but which about a century ago took on an entirely new aspect, thanks to the ideas of Lobachevsky and of Bolyai. For a long time we attempted to demonstrate the proposition known as the postulate of Euclid; we constantly failed; we know now the reason for these failures. Lobachevsky succeeded in building a logical edifice as coherent as the geometry of Euclid, but in which the famous postulate is assumed false, and in which the sum of the angles of a triangle is always less than two right angles. Riemann devised another logical system, equally free from contradiction, in which this sum is on the other hand always greater than two right angles. These two geometries, that of Lobachevsky and that of Riemann, are what are called the non-Euclidean geometries. The postulate of Euclid then cannot be demonstrated; and this impossibility is as absolutely certain as any mathematical truth whatsoever - a fact which does not prevent the Acad6mie des Sciences from receiving every year several new proofs, to which it naturally refuses the hospitality of the Comptes rendus. Much has already been written on the non-Euclidean geometries; once they scandalized us; now we have become accustomed to their paradoxes; some people have gone so far as to doubt the truth of the postulate and to ask whether real space is plane, as Euclid assumed, or whether it may not present a slight curvature. They even supposed that experiment could give them an answer [250] to this question. Needless to add that this was a total misconception of the nature of geometry, which is not an experimental science. But why, among all the axioms of geometry, should this postulate be the only one which could be denied without offence to logic? Whence should it derive this privilege? There seems to be no good reason for this, and many other conceptions are possible. However, many contemporary geometers do not appear to think so. In recognizing the claims of the two new geometries they feel doubt147 P. Ehrlich (ed.), Real Numbers, Generalizationsof the Reals, and Theories of Continua, 147-168. © 1994 Kluwer Academic Publishers. Printed in the Netherlands.

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less that they have gone to the extreme limit of possible concessions. It is for this reason that they have conceived what they call general geometry, which includes as special cases the three systems of Euclid, Lobachevsky, and Riemann, and does not include any other. And this term general indicates clearly that, in their minds, no other geometry is conceivable. They will lose this illusion if they read the work of Professor Hilbert. In it they will find the barriers behind which they have wished to confine us broken down at every point. To understand well this new attempt we must recall what has been the evolution of mathematical thought for the last hundred years, not only in geometry, but in arithmetic and in analysis. The concept of number has been made more clear and precise; at the same time it has been generalized in various directions. The most valuable of these generalizations for the analyst is the introduction of imaginaries which the modern mathematician could not now dispense with; but we have not stopped with this; other generalizations of number, or, as we say, other categories of complex numbers, have been introduced into science. The operations of arithmetic have in their turn been subjected to criticism, and Hamilton's quaternions have given us an example of an operation which presents an almost perfect analogy to multiplication, and may be called by the same name, which, however, is not commutative, that is, the product of two factors is not the same when the order of the factors is reversed. This was a revolution in arithmetic quite comparable to that which Lobachevsky effected in geometry. Our conception of the infinite has been likewise modified [251]. Professor G. Cantor has taught us to distinguish gradations in infinity itself (which have, however, nothing to do with the infinitesimals of different orders invented by Leibniz for the ordinary infinitesimal calculus). The concept of the continuum, long regarded as a primitive concept, has been analyzed and reduced to its elements. Shall I mention also the work of the Italians, who have endeavored to construct a universal logical symbolism and to reduce mathematical reasoning to purely mechanical rules? We must recall all this if we wish to understand how it is possible that conceptions which would have staggered Lobachevsky himself, revolutionary as he was, can seem to us to-day almost natural, and can be propounded by Professor Hilbert with perfect equanimity.

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THE LIST OF AXIOMS

The first thing to do was to enumerate all the axioms of geometry. This was not so easy as one might suppose; there are the axioms which one sees and those which one does not see, which are introduced unconsciously and without being noticed. Euclid himself, whom we suppose an impeccable logician, frequently applies axioms which he does not expressly state. Is the list of Professor Hilbert final? We may take it to be so, for it seems to have been drawn up with care. The distinguished professor divides the axioms into five groups: I. Axiome der Verkniipfung (I shall translate by projective axioms [axiomes projectifs] instead of trying to find a literal translation, as for example axioms of connection [axiomes de la connection], which would not be satisfactory). II. Axiome der Anordnung (axioms of order [axiomes de l'ordre]). III. Axiome of Euclid. IV. Axioms of congruence or metrical axioms. V. Axiom of Archimedes. Among the projective axioms, we shall distinguish those of the plane and those of space; the first are those derived from the familiar proposition: through two points passes one and only one straight line; - but I prefer to translate literally, in order to make Professor Hilbert's thought well understood. Let us suppose three systems of objects which we shall call points [252), straight lines, and planes. Let us suppose that these points, straight lines, and planes are connected by certain relations which we shall express by the words lying on, between, etc.

I. - 1. Two different points A and B determine always a straight line a; in notation AB = a

or

BA = a.

In place of the word determine we shall employ as well other turns of phrase which shall be synonymous; we shall say: A lies on a, A is a point of a, a passes through A, a joins A and B, etc. 1. - 2. Any two points of a straight line determine this straight line; that is, if AB = a and AC = a, and if B is different from C, we have also BC = a.

The following are the considerations which these statements are intended to suggest: the expressions lying on, passing through, etc., are not meant to call up mental pictures; they are simply synonyms of the word determine. The words point, straight line, and plane themselves are not intended to arouse in the mind, any visual image representationn

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sensible]. They might denote indifferently objects of any sort whatever, provided one could establish among these objects a correspondence such that to every pair of the objects called points there would correspond one and only one of the objects called straight line. And this is why it becomes necessary to add (I, 2) that, if the line which corresponds to the pair of point A and B is the same as that which corresponds to the pair of points B and C, it is also the same as that which corresponds to the pair of points A and C. Thus Professor Hilbert has, so to speak, sought to put the axioms into such a form that they might be applied by a person who would not understand their meaning because he had never seen either point or straight line or plane. It should be possible, according to him, to reduce reasoning to purely mechanical rules, and it should suffice, in order to create geometry, to apply these rules slavishly to the axioms without knowing what the axioms mean. We shall thus be able to construct all geometry, I will not say precisely without understanding it at all, since we shall grasp the logical connection of the [253] propositions, but at any rate without seeing it at all. We might put the axioms into a reasoning apparatus like the logical machine1 of Stanley Jevons, and see all geometry come out of it. This is the same consideration that has inspired certain Italian scholars, such as Peano and Padoa, who have endeavored to create a pasigraphy, that is, a sort of universal algebra, where all the processes of reasoning are replaced by symbols or formulas. This notion may seem artificial and puerile; and it is needless to point out how disastrous it would be in teaching and how hurtful to mental development; how deadening it would be for investigators, whose originality it would nip in the bud. But, as used by Professor Hilbert, it explains and justifies itself, if one remembers the end pursued. Is the list of axioms complete, or have we overlooked some which we apply unconsciously? This is what we want to know. For this we have one criterion, and only one. We must find out whether geometry is a logical consequence of the axioms explicitly stated, that is, whether, if we put these axioms into the reasoning machine, we can make the whole sequence of propositions come out. If we can, we shall be sure that nothing has been overlooked. For our machine cannot work except according to the rules of logic for which it has been constructed; it ignores the vague instinct which we call intuition.

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I shall not enlarge upon the projective axioms of space, which the author numbers 1, 3, 4, 5, 6. Nothing is changed from the usual statements. A word only on the axiom I, 7, which is thus formulated: On every straight line there are at least two points; on every plane there are at least three points not in a straight line; in space there are at least four points which are not in the same plane.

This statement is characteristic. Any one who had left any place for intuition, however small it might be, would not have dreamed of saying that on every straight line there are at least two points, or rather he would have added at once that there are an infinite number of them; for the intuition of the [254] straight line would have revealed to him both facts immediately and simultaneously. Let us pass to the second group, that of the axioms of order. Here is the statement of the first two: If three points are on the same straight line, there is a certain relation among them which

we express by saying that one of the points, and only one, is between the other two. If C is between A and B, and D between A and C, then D will be also between A and B, etc.

Here again we do not bring in our intuition; we are not seeking to fathom what the word between may signify; every relation which satisfies the axioms might be denoted by the same word. This is an illuminating example of the purely formal nature of mathematical definitions; but I do not dwell upon it, since I should have simply to repeat what I have said already, in speaking of the first group. But another consideration forces itself upon us. The axioms of order are presented as dependent on the projective axioms, and they would not have any meaning if we did not admit these latter, since we should not know what are three points on a straight line. And nevertheless there exists a special geometry, purely qualitative, which is entirely independent of projective geometry, and does not assume the idea of the straight line, nor that of the plane, but only the ideas of curves and surfaces; this is what is called analysis situs. Would it not be preferable to give to the axioms of the second group a form which would free them from this dependence and separate them completely from the first group? It remains to be seen whether this would be possible, while preserving the purely logical character of these axioms, that is, while closing the door completely against all intuition. The third group contains only a single axiom, which is the famous

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postulate of Euclid; I shall note simply that, contrary to the usual custom, it is presented before the metrical axioms. These last form the fourth group. We shall divide them into three subgroups. The propositions IV, 1, 2, 3 are the metrical axioms for segments: these axioms serve to define length. We shall agree to say that a segment taken on a [255] straight line may be congruent (equal) to a segment taken on another straight line; this is axiom IV, 1; but this convention is not wholly arbitrary; it must be so made that two segments congruent to the same third segment shall be congruent to each other (IV, 2). In the next place we define the addition of segments, by a new convention; and this convention, in turn, must be so made that when we add equal segments we find the sums equal; and this is axiom IV, 3. The propositions IV, 4, 5 are the corresponding axioms for angles. But these are not yet sufficient; to the two subgroups of metrical axioms for segments and for angles we must add the metrical axiom for triangles (which Professor Hilbert numbers IV, 6): if two triangles have an equal angle included between equal sides, the other angles of these two triangles are equal each to each. We recognize here one of the well known cases of equality of triangles, which we usually demonstrate by superposition, but which we must set up as a postulate if we wish to avoid making appeal to intuition. Moreover, when we made use of intuition, that is of superposition, we saw by the same process that the third sides were equal in the two triangles, and these two propositions were united, so to speak, in a single apperception; here, on the contrary, we separate them; one of them we make a postulate, but we do not set up the other as a postulate, since it can be logically deduced from the first. Another comment: Professor Hilbert says distinctly that the segment AB is congruent to itself, but (and the same is true for angles) he should have added, should he not, that it is congruent to the inverse segment BA. This axiom (which implies the symmetry of space) is not identical with those which are explicitly stated. I do not know whether it could be logically deduced from them; I believe it could, but, given the course of reasoning of Professor Hilbert, it seems to me that this postulate is applied without being stated (page 17, line 18). I also regret that, in this exposition of the metrical axioms, there remains no trace of an idea whose importance Helmholtz was the first to understand: I refer to the displacement of a rigid figure. It would

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have been possible to preserve this idea in its nature r6le, without sacrificing the logical character of the axioms. One might have said, for example: I define between figures a [256] certain relations which I call congruence, etc.; two figures which are congruent to the same third figure are congruent to each other; two congruent figures are identical when three points of one, not in a straight line, are identical with three corresponding points of the other, etc. The artificial introduction of this axiom IV, 6 would thus have been avoided, and the postulates would have been brought into connection with their actual psychological origin. The fifth group contains only a single axiom, that of Archimedes. Let A and B be any two points on a straight line D; let a be any segment; starting from the point A, and in the direction AB, construct on D a series of segments, all equal to each other and equal to a: AA 1 , A1A 2... A ._,A,; then we shall always be able to take n so great that the point B will be found on one of these segments. That is to say, if we have given two lengths 1 and L, we can always find a whole number n so great that when we add the length I to itself n times, we obtain a total length greater than L. INDEPENDENCE OF THE AXIOMS

The list of axioms once drawn up, we must see whether it is free from contradiction. We know well that it is, since geometry exists; and Professor Hilbert also answers in the affirmative, by constructing a geometry. But this geometry, strange to say, is not quite the same as ours, his space is not our space, or at least is only a part of it. In the space of Professor Hilbert we do not have all the points which there are in our space, but only those which we can construct by ruler and compass, starting from two given points. In this space, for example, there would not exist, in general, an angle which would be the third part of a given angle. I have no doubt that this conception would have been regarded by Euclid as more rational than ours. At any rate it is not ours. To come back to our geometry it would be necessary to add an axiom: If, on a straight line, there is a double infinity of points A1, A2, .... A, ... ; B 1, B2, S.... B, B...., such that Bq is included between Ap and [2571 Bq-,, and A. between Bq and Ap_1 , whatever the values of p and q, then there will be on this straight line at least one point C which lies between AP and Bq, whatever the values of p and q.

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We must ask next whether the axioms are independent, that is, whether we could sacrifice one of the five groups, retaining the other four, and still attain a coherent geometry. Thus by suppressing group III (the postulate of Euclid), we obtain the non-Euclidean geometry of Lobachevsky. In the same way, we can suppress group IV. Professor Hilbert has succeeded in retaining groups I, II, III and V, along with the two subgroups of metrical axioms for segments and for angles, while rejecting the metrical axiom for triangles, that is, proposition IV, 6. This is how he accomplishes it: consider, for simplicity, plane geometry, and let P be the plane in which we operate; we shall retain the usual meaning for the words point and straightline, and also the usual measurement of angles; but not so for lengths. A length shall be measured by definition by its projection on a plane Q different from P, this projection itself being measured in the usual way. It is clear that all the axioms will hold, except the metrical axioms. The metrical axioms for angles will also hold, since we change nothing concerning the measurement of angles; those for segments will also hold, since each segment is measured by another segment which is its projection on the plane Q, and this latter segment is measured in the usual way. On the other hand, the theorems on the equality of triangles, such as the axiom IV, 6, are no longer true. This solution satisfies me only half-way; angles have been defined independently of lengths, without trying to bring the two definition into agreement (or rather, by bringing them purposely into disagreement). To return to classic geometry it would be sufficient to change one of the two definitions. I should prefer to have had the lengths so defined as to make it impossible to find a definition of angles satisfying the metrical axioms for angles and for triangles. This would moreover not be difficult [258]. It would have been easy for Professor Hilbert to create a geometry in which the axioms of order would be abandoned while all the others would be retained. Or rather this geometry exists already, or rather there exist two of them. There is that of Riemann, for which, it is true, the postulate of Euclid (group III) is also abandoned, since the sum of the angles of a triangle of greater than two right angles. To make my thought clear I shall limit myself to considering a geometry of two dimensions. The geometry of Riemann in two dimensions is nothing else than spherical geometry, with one condition, namely, that we shall not regard as distinct two diametrically opposite points on the sphere. The elements

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of this geometry will then be the different diameters of this sphere. Now, if we consider three diameters of the same sphere, lying in the same diametral plane, we have no reason for saying that one of them is between the other two. The word between has no longer any meaning, and the axioms of order drop out of themselves. If we wish now a geometry in which the axioms of order shall not hold, while the axiom of Euclid is retained with the others, we have only to take as elements the imaginary points and straight lines in ordinary space. It is clear that the imaginary points of space are not given us as arranged in a definite order. But more than that: we may ask whether they are capable of being so arranged; this would undoubtedly be possible, as G. Cantor has shown (subject to the condition, be it understood, of not always arranging in close proximity points which we regard as infinitely near and of destroying thereby the continuity of space). We might, I say, arrange them, but this could not be done in such a way that the arrangement would not be altered by the various operations of geometry (projection, translation, rotation, etc.) The axioms of order, then, are not applicable to this geometry. THE NON-ARCHIMEDEAN

GEOMETRY

But the most original conception of Professor Hilbert is that of nonArchimedean geometry, in which all the axioms remain true except that of Archimedes. For this it was necessary, in the first place, to construct a system [259] of non-Archimedean numbers, that is, a system of elements among which we may define the relations of equality and inequality and to which we may apply operations analogous to arithmetical addition and multiplication - and this in such a way as to satisfy the following conditions: 10 The arithmetical rules for addition and multiplication (the commutative, associative, distributive laws, etc.: Arithmetische Axiome der Verkniipfung) hold without change. 2' The rules for the establishment and transformation of inequalities (Arithmetische Axiome der Anordnung) likewise hold. 30 The axiom of Archimedes is not true. We may reach this result by choosing for elements series of the following form:

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where m is a positive or negative integer and where the coefficients A are real, and by agreeing to apply to these series the ordinary rules of addition and multiplication. We must then define the conditions of inequality of these series, so as to arrange our elements in a definite order. We shall accomplish this by the following convention: we shall give to our series the sign of A0 and we shall say that one series is less than another when, subtracted from the other, it leaves a positive remainder. It is clear that with this convention the rules of the calculus of inequalities hold; but the axiom of Archimedes is no longer true; for, if we take the two elements I and t, the first added to itself as many times as we please remains always less than the second. We shall have always t > n, whatever the whole number n, since the difference t - n will always be positive; for the coefficient of the first term t, which, by definition, gives its sign, remains always equal to 1. Our ordinary numbers come in as particular cases among these nonArchimedean numbers. The new numbers are interpolated, so to speak, in the series of our ordinary numbers, in such a way that we may have, for example, as infinity of the new numbers less than a given ordinary number A and greater than all the ordinary numbers less than A [260]. This premised, imagine a space of three dimensions in which the coordinates of a point would be measured not by ordinary numbers but by non-Archimedean numbers, while the usual equations of the straight line and the plane would hold, as well as the analytic expressions for angles and lengths. It is clear that in this space all the axioms would remain true except that of Archimedes. On every straight line new points would be interpolated between our ordinary points. If, for example, Do is an ordinary straight line, and D, the corresponding non-Archimedean straight line; if P is any ordinary point of Do, and if this point divides Do into two half-rays S and S' (I add, for precision, that I consider P as not belonging to either S or S'); then there will be on D1 an infinity of new points as well between P and S as between P and S'. There will be also on D, an infinity of new points which will lie to the right of all the ordinary points of D,. In short, our ordinary space is only a part of the non-Archimedean space. At the first blush the mind revolts against conceptions like this. This is because, through an old habit, it is looking for a visual image. It must free itself from this prejudice if it would arrive at comprehension, and this is even more necessary here than in the case of non-Euclidean

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geometry. Professor Hilbert has only one object in view: to construct a system of elements capable of certain logical relations; and it is sufficient for him to show that these relations do not involve any selfcontradiction. We may remark in passing that the non-Euclidean geometry respects, so to speak, our qualitative conception of the geometrical continuum, while entirely overturning our ideas about the measurement of this continuum. The non-Archimedean geometry destroys this concept, by dissecting the continuum for the introduction of new elements. Whatever they may be, Professor Hilbert follows out the consequences of his premises and tries to see how one could remake geometry without using the axiom of Archimedes. There is no difficulty in the chapters which the school-boys call the first and second Books. This axiom does not occur at any point in those Books. The third Book treats of proportions and of similarity. The plan which Professor Hilbert follows for the [261] reconstruction of this book without recourse to the axiom of Archimedes is, in substance, as follows. He takes the usual construction of the fourth proportional as the definition of proportion; but such a definition needs to be justified; he needs to show in the first place that the result is the same whatever may be the auxiliary lines employed in the construction, and next that the ordinary rules or operation apply to the proportions thus defined. This justification Professor Hilbert gives us in a satisfactory manner. The fourth Book treats of the measurement of plane areas. If this measurement can be easily established without the aid of the principle of Archimedes, it is because two equivalent polygons can either be decomposed into triangles in such a way that the component triangles of the one and those of the other are equal each to each (or, in other words, can be converted one into the other after the manner of the Chinese puzzle 2), or else can be regarded as the difference of polygons capable of this mode of decomposition (this is really the same process, admitting not only positive triangles but also negative triangles). But we must observe that an analogous state of affairs does not seem to exist in the case of two equivalent polyhedra, so that it becomes a question whether we can determine the volume of the pyramid, for example, without an appeal more or less disguised to the infinitesimal calculus. It is then not certain whether we could dispense with the axiom of Archimedes as easily in the measurement of volumes as in that of plane areas. Moreover Professor Hilbert has not attempted it.

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One question remains to be treated in any case; a polygon being given, is it possible to cut it up into triangles and remove one of the pieces in such a way that the remaining polygon may be equivalent to the given polygon, that is to say, in such a way that by transforming this remaining polygon by the process of the Chinese puzzle we could come back to the original polygon? Ordinarily we are satisfied with saying that this is impossible because the whole is greater than the part. This is to call in a new axiom, and, however obvious it may seem to us, the logician would be better satisfied if we could avoid it. Professor Schur has discovered a proof, it is true, but it depends on the axiom of Archimedes; Professor Hilbert wished to reach the result without using this axiom. This is the device by which he [262] does it: he adopts as the definition of the area of the triangle half the product of its based by its altitude, and he justifies this definition by showing that two triangles which are equivalent (from the point of view of the Chinese puzzle) have the same area (in the sense of the new definition) and that the area of a triangle which can be decomposed into several others is the sum of the areas of the component triangles. This justification once out of the way, all the rest follows without difficulty. It is always the same process. To avoid constant appeals to intuition, which would provide us constantly with new axioms, we change these axioms into definitions, and afterwards justify these definitions by showing that they are free from contradictions. THE NON-ARGUESIAN GEOMETRY

The fundamental theorem of projective geometry is the theorem of Desargues. Two triangles are called homologous when the straight lines which join the corresponding vertices intersect in the same point. Desargues has shown that the points of intersection of the corresponding sides of two homologous triangles are on the same straight line; the converse is also true. The theorem of Desargues can be established in two ways: 10 By using the projective axioms of the plane and the metrical axioms of the plane. 20 By using the projective axioms of the plane and those of space. The theorem might then be discovered by a two-dimensional animal, to whom a third dimension would seem as inconceivable as a fourth does to us; such an animal would then be ignorant of the projective axioms

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of space; but he would have seen movement, in the plane which he inhabits, of rigid figures analogous to our rigid bodies, and would consequently be acquainted with the metrical axioms. The theorem could be discovered also by a three-dimensional animal who was acquainted with the projective axioms of space, but who, never having seen rigid bodies move, would be ignorant of the metrical axioms. But could we establish the theorem of Desargues without using either the projective axioms of space or the metrical axioms, [263] but only the projective axioms of the plane? We thought not, but we were not sure of it. Professor Hilbert has decided the question by constructing a nonArguesian geometry, which is, of course, a plane geometry. Consider an ellipse E. Outside of this ellipse the word straight line preserves its ordinary meaning: in the interior the word straight line takes a difference meaning and denotes an arc of a circle which, when produced, would pass through a fixed point P out-side the ellipse. A straight line which crosses the ellipse E is then composed of two rectilinear parts, in the ordinary sense of the word, connected in the interior of the ellipse by an arc of a circle; like a ray of light which would be deflected from its rectilinear path by passing through a refracting body. The projective axioms of the plane will still be true if we take the point P sufficiently far removed from the ellipse E. Now place two homologous triangles outside the ellipse E, and in such a way that their sides do not meet E; the three straight lines which join the corresponding vertices two and two, if we take them in the ordinary sense of the word, will meet in the same point Q, according to the theorem of Desargues; suppose that this point Q is in the interior of E. If we take the word straight line in the new sense, the three straight lines which join the corresponding vertices will be deflected on entering the interior of the ellipse. They will then no longer pass through Q, they will be no longer concurrent. The theorem of Desargues is no longer true in our new geometry; this is a non-Arguesian geometry. THE NON-PASCALIAN

GEOMETRY

Professor Hilbert does not stop here, but introduces still another new conception. In order to understand it, we must first return a moment into the domain of arithmetic. We have noticed above the extension of the concept of number, by the introduction of the non-Archimedean numbers. We want a classification of these new numbers, to obtain which

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we shall begin by classifying the axioms of arithemetic in four groups, which are: 10 The associative and commutative laws of addition, the associative law of multiplication, the two [264] distributive laws of multiplication; or, in short, all the rules of addition and of multiplication, except the commutative law of multiplication; 20 The axioms of order; that is, the rules of the calculus of inequalities; 30 The commutative law of multiplication, according to which we can invert the order of the factors without changing the product; 40 The axiom of Archimedes. Numbers which admit the axioms of the first two groups shall be called Arguesian; they may be Pascalianor non-Pascalian,according as they satisfy or do not satisfy the axiom of the third group; they will be Archimedean or non-Archimedean, according as they satisfy or not the axiom of the fourth group. We shall soon see the reason for these names. Ordinary numbers are at the same time Arguesian, Pascalian and Archimedean. It can be shown that the commutative law follows from the axioms of the first two groups and the axiom of Archimedes; there are therefore no numbers which are Arguesian, Archimedean and not Pascalian. On the other hand, we have cited above an example of numbers which were Arguesian, Pascalian and not Archimedean; I shall call these the numbers of the system T, and I recall that to each of these numbers there corresponds a series of the form Aotm + Altm-I + . ... , where the As are ordinary real numbers. It is easy to construct, by an analogous process, a system of Arguesian numbers which are non-Pascalian and non-Archimedean. The elements of this system will be series of the form S = Tosn + Ts n -1

+ ...

,

where s is a symbol analogous to t, n a positive or negative integer, and To, T1, . . . numbers of the system T; if then we replaced the coefficients To, T1, . .. by the corresponding series in t we should have a series depending on both t and s. We shall [265] add these series S according to the ordinary rules; also for the multiplication of these series we shall admit the distributive and associative laws; but we shall

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suppose that the commutative law is not true, that on the contrary st = -ts. It remains to arrange the series in a definite order, so as to satisfy the axioms of order. For this, we shall attribute to the series S the sign of the first coefficient TO; we shall say that one series is less than another when subtracted from the first, it leaves a positive remainder. It is always the same scheme: t is regarded as very great in comparison with any ordinary real number, and s is regarded as very great in comparison with any number of the system T. The commutative law not being true, these are clearly non-Pascalian numbers. Before going farther, I recall that Hamilton introduced long ago a system of complex numbers in which multiplication is not commutative; these are the quaternions,of which the English make such frequent use in mathematical physics. But, in the case of quaternions, the axioms of order are not true; the originality of Professor Hilbert's conception lies in this, that his new numbers satisfy the axioms of order without satisfying the commutative law. To return to geometry. Admit the axioms of the first three groups, that is, the projective axioms of the plane and of space, the axioms of order and the postulate of Euclid: the theorem of Desargues will follow from them, since it is a consequence of the projective axioms of space. We wish to construct our geometry without making use of the metrical axioms; the word length has then for us no meaning; we have no right to use the compass; on the other hand, we may use the ruler, since we admit that we can draw a straight line through two points, by virtue of one of the projective axioms; also, we know how to draw through a given point a parallel to a given straight line, since we admit the postulate of Euclid. Let us see what we can do with these resources. We can define the homothetic relation3 [homothitie] of two figures; two triangles shall be called homothetic when their sides are [266] parallel two and two, and we conclude from this (by the theorem of Desargues, which we admit) that the straight lines which join the corresponding vertices are concurrent. We shall then make use uf the homothetic relation to define proportion. We can also define equality to a certain extent. Two opposite sides of a parallelogram shall be equal by definition; we can thus decide whether two segments are equal to each other, provided they are parallel. Thanks to these conventions, we are now in a position to compare

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the lengths of two segments; provided, however, that these segments are PARALLEL. The comparison of two lengths which have different directions has no meaning; there would be required, so to speak, a different unit of length for each direction. Needless to add that the word angle has no meaning. Lengths will thus be expressed by numbers; but these will not necessarily be ordinary numbers. All that we can say is this, that, if the theorem of Desargues is true, as we admit, these numbers will belong to a system satisfying the arithmetic axioms of the first two groups, that is, to an Arguesian system. Conversely, being given any system S of Arguesian numbers, we can construct a geometry in which the lengths of segments of a straight line can be exactly expressed by these numbers. Here is the way in which this can be done: a point of this new space shall be defined by three numbers x, y, z of the system S which we shall call the coordinates of this point. If to the three coordinates of the various points of a figure we add three constants (which are, of course, Arguesian numbers of the system S), we obtain another figure, derived from the first in such a way that to any segment of one of the figures there corresponds and equal and parallel segment in the other (in the sense given above to this word). This transformation is then a translation, so that these three constants might define a translation. If now we multiply the three coordinates of all the points of a given figure by the same constant, we shall obtain a second figures which will be homothetic to the first [267]. The equation of a plane will be the well known linear equation of ordinary analytic geometry; but, since in the system S multiplication will not in general be commutative, it is important to make a distinction and to say that in each of the terms of this linear equation that factor shall be the coordinate which plays the r6le of multiplicand, and that the constant coefficient which plays the r6le of multiplier. Thus, to each system of Arguesian numbers there will correspond a new geometry satisfying the projective axioms, the axioms of order, the theorem of Desargues, and the postulate of Euclid. What is now the geometric meaning of the arithmetic axiom of the third group, that is, of the commutative law of multiplication? Translated into geometric language, this law is the theorem of Pascal; I refer to the theorem on the hexagon inscribed in a conic, supposing that this conic reduces to two straight lined.

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Thus the theorem of Pascal will be true or false according as the system S is Pascalian or non-Pascalian; and, since there are non-Pascalian systems, there are also non-Pascaliangeometries. The theorem of Pascal can be proved by starting with the metrical axioms; it will then be true, if we admit that figures can be transformed not only by the homothetical transformation and translation, as we have just been doing, but also by rotation. The theorem of Pascal can also be deduced from the axiom of Archimedes, since we have just seen that every system of numbers which is Arguesian and Archimedean is at the same time Pascalian; every non-Pascaliangeometry is then at the same time non-Archimedean. The Streckeniibertrager Let us mention one more idea of Professor Hilbert. He studies the constructions which can be made, not with the aid of the ruler and compass, but by means of the ruler and a special instrument which he calls Streckeniibertrager,and which would enable us to lay off on a straight line a segment equal to another segment taken on another straight line. The Streckeniibertrageris not equivalent to the compasses; this latter instrument would enable us to construct the point of intersection of two circles or of a [268] circle and any straight line; the Streckeniibertrager would give us only the intersection of a circle with a straight line passing through the center of the circle. Professor Hilbert inquires then what constructions will be possible with these two instruments, and he reaches a quite remarkable conclusion. The constructions which can be made by the ruler and compass can also be made by the ruler and the Streckeniibertrager,if these constructions are such that their result is always real. It is easy enough to see that this condition is necessary; for a circle is always cut in two real points by a straight line drawn through its centre. But it was difficult to foresee that this condition would be also sufficient. Various Geometries I should like, before closing, to see what places are taken in Professor Hilbert's classification by the various geometries which have been proposed up to the present time. In the first place, the geometries of Riemann; I do not mean the geometry of Riemann which has been

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mentioned above and which is contrasted with that of Lobachevsky; I mean the geometries connected with space of variable curvature considered by Riemann in his celebrated Habilitationsschrift. In this conception, any curve has a length assigned to it, by definition, and it is on this definition that everything depends. The role of straight lines is played by the geodesics, that is, by the lines of minimum length drawn from one point to another. The projective axioms are no longer true, and there is no reason why, for example, two point could not be joined by more than one geodesic. The postulate of Euclid clearly can no longer have any meaning. The axiom of Archimedes remains true, as well as the axioms of order mutatis mutandis; Rienmann does not consider, indeed, any but the ordinary system of numbers. As to the metrical axioms, it is easily seen that those for segments and those for angles remain true, while the metrical axiom for triangles (IV, 6) is evidently false. And here we meet the objection which has been most often made to Riemann. You speak of length, [they say to him] now length assumes measurement, and to measure we must be able to carry about a measuring [2691 instrument which must remain invariant; moreover, you recognize this yourself. Space then must be everywhere equal to itself, it

must be homogeneous in order that congruence may be possible. Now your space is not homogeneous, since its curvature is variable; in such a space there can be no such thing

as measurement of length. Riemann would have had no trouble in replying. Consider, for simplicity, a geometry of two dimensions; we shall be able then to picture to ourselves Riemann's space as a surface in ordinary space. We might measure lengths on this surface by means of a thread, and nevertheless a figure could not be moved about in this surface in such a way that the lengths of all its elements remain invariant. For the surface is not, in general, applicable on itself. This is what Professor Hilbert would express by saying that the metrical axioms for segments are true, while that for triangles is not. The first find concrete expression, so to speak, in our thread; the axiom for triangles would assume a displacement of a figure all of whose elements would have a constant length. What will be the place of another geometry which I have proposed on a former occasion4 and which belongs, so to speak, to the same family as that of Lobachevsky and that of Riemann? I have shown that we can imagine three geometries in two dimensions, which correspond

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respectively to three kinds of surfaces of the second degree: the ellipsoid, the hyperboloid of two sheets, and the hyperboloid of one sheet; the first is that of Riemann, the second is that of Lobachevsky, and the third is the new geometry. We should find in the same way four geometries in three dimensions. Where would this new geometry stand in the classification of Professor Hilbert? It is easy to discover. As in the case of the geometry of Riemann, all the axioms hold, save those of order and that of Euclid; but, while in the geometry of Riemann the axioms are false on all the straight lines, in the new geometry, on the contrary, the straight lines separate themselves into two classes, those on which the axioms of order are true, and those on which they are false [270]. Conclusions But the most important thing is to arrive at a clear understanding of the place which the new conceptions of Professor Hilbert occupy in the history of our ideas on the philosophy of mathematics. After a first period of naive confidence in which we cherished the hope of demonstrating everything, came Lobachevsky, the inventor of the non-Euclidean geometries. But the true meaning of this discovery was not fathomed all at once; Helmholtz showed in the first place that the propositions of Euclidean geometry were no other than the laws of motion of rigid bodies, while the propositions of the other geometries were the laws which might govern other bodies analogous to the rigid bodies - bodies which doubtless do not exist, but whose existence might be conceived without leading to the least contradiction, bodies which we might fabricate if we wished. These laws could not, however, be regarded as experimental, since the solids of nature follow them only roughly, and since, besides, the fictitious bodies of non-Euclidean geometry do not exist, and cannot be accessible to experiment. Helmholtz, moreover, never explained himself altogether clearly on this point. Lie pushed the analysis much farther. He inquired in what way the various possible movements of any system, or more generally the various possible transformations of a figure, can be combined. If we consider a certain number of transformations, and suppose that they are combined in all possible ways, the totality of all these combinations will form what he calls a group. To each group corresponds a geometry, and ours,

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which corresponds to the group of displacements of a rigid body, is only a very special case. But all the groups which one can imagine will possess certain common properties, and it is precisely these common properties which limit the caprice of the inventors of geometries; it is they, indeed, which Lie studied all his life. He was, however, not entirely satisfied with his work. He had, he said, always regarded space as a Zahlenmannigfaltigkeit.He had confined himself to the study of continuous groups [271] properly so called, to which the rules of the ordinary infinitesimal analysis apply. Was he not thus artificially restricted? Had he not thus neglected one of the indispensable axioms of geometry (referring to the axiom of Archimedes)? I do not know whether any trace of this thought would be found in his printed works, but in his correspondence, or in his conversation, he constantly expressed this same concern. This is precisely the gap which Professor Hilbert has filled up; the geometries of Lie remained all subject to the forms of analysis and of arithmetic, which seemed unassailable. Professor Hilbert has broken through these forms, or, if you prefer, he has enlarged them. His spaces are no longer Zahlenmannigfaltigkeiten. The objects which he calls points, straight lines, or planes become thus purely logical entities which it is impossible to represent to ourselves. We should not know how to picture them as sensory images, these points which are nothing but systems of three series. It matters little to him; it is sufficient for him that they are individuals and that he has positive rules for distinguishing these individuals one from another, for establishing arbitrarily between them relations of equality or of inequality, and for transforming them. One other comment: the groups of transformations in Lie's sense appear to play only a secondary part. At least this is how it seems when we read the actual text of Professor Hilbert. But, if we should consider it more closely, we should see that each of his geometries is still the study of a group. His non-Archimedean geometry is the study of a group which contains all the transformations of the Euclidean group, corresponding to the various displacements of a rigid body, but which contains also other transformations capable of being combined with the first according to simple laws. Lobachevsky and Riemann rejected the postulate of Euclid, but they preserved the metrical axioms; in the majority of his geometries,

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Professor Hilbert does the opposite. This amounts to placing in the first

rank a group comprising the transformations of space by the homothetic transformation and by translation; and at the foundation of his

non-Pascalian geometry we meet an analogous group, comprising not only the homothetic transformation and the translations of ordinary space, but other analogous transformations which combine with the first according to simple laws [272].

Professor Hilbert seems rather to slur over these inter-relations; I do not know why. The logical point of view alone appears to interest him.

Being given a sequence of propositions, he finds that all follow logically from the first. With the foundation of this first proposition, with

its psychological origin, he does not concern himself. And even if we have, for example, three propositions A, B, C, and if it is logically possible, by starting with any one among them, to deduce the other two from it, it will be immaterial to him whether we regard A as an axiom, and derive B and C from it, or whether, on the contrary, we regard C as an axiom, and derive A and B from it. The axioms are postulated;

we do not know where they come from; it is then as easy to postulate A as C.

His work is then incomplete; but this is not a criticism which I make against him. Incomplete one must indeed resign one's self to be. It is

enough that he has made the philosophy of mathematics take a long step in advance, comparable to those which were due to Lobachevsky,

to Riemann, to Helmholtz, and to Lie. NOTES Translated, with the author's permission, by Dr. E. V. Huntingston. The original review appeared in Darboux's Bulletin des Sciences Mathimatiques, 2d ser., vol. 26 (September, 1902), pp. 249-272, and also, with some modification of the more technical passages, in the Journaldes Savants for 1902 (May), pp. 252-271. The present translation (except the postscript) is from the version in Darboux's Bulletin, the heavy faced figures in brackets indicating the pages of the original; the postscript appeared only in the Journaldes Savants (p. 271). The Grundlagen der Geometrie, by Professor David Hilbert (Leipzig, Taubner, 8vo, 92 pp.), appeared in 1899, and was reviewed for the Bulletin by Dr. Sommer (vol. 6, 1900, pp. 287-299). A French translation by Professor L. Laugel and an English translation by Professor E. J. Townsend appeared in 1900 and 1902, and were reviewed by Dr. E. R. Hedrick in the Bulletin, vol. 9 (1902), pp. 158-165. See also a review by Mr. 0. Veblen in the Monist, vol. 13 (January, 1903), pp. 303-309. *

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This Review (including a postscript which has been omitted) first appeared in the Bulletin of the American Mathematical Society, Series 2, 10, 1-23 (1903). We are grateful to the AMS for permission to publish it here. SCf. Lond. Phil. Trans., vol. 160 (1870), pp. 497-518 [Translator]. 2 By cutting up and putting together again [Translator]. Two figures are homothetic when they are similar and similarly placed [Translator]. See Bull. de la Socigtg mathimatique de France, vol. 15 (1887), pp. 203-216. Other articles by Poincar6 on the foundations of geometry have appeared in the Revue de Mitaphysique, vol. 7, and the Monist, vol. 9 [Translator].

GIUSEPPE VERONESE

ON NON-ARCHIMEDEAN GEOMETRY"1

2

The subject I have chosen is that which I was asked to discuss at the Congress of Heidelberg; 3 it seems to me that it can still be of interest to you today since mathematicians such as Poincar6 have recognised its importance.4 Critics have already recognized its logical validity; therefore, instead of attempting a systematic exposition, as I would have done in Heildelberg, I believe that it is more valuable to focus here on the questions of content and method which are connected to the essence of the principles of pure mathematics and of geometry, upon which it seems to me that geometers have not yet agreed, although these are questions of geometry.5 What is non-Archimedean geometry? Is it worthy as a system of abstract truths? Does it also satisfy the conditions to which any geometrical system must be submitted? It would be of no interest to recall here the age-old debates on the actual infinite and infinitesimal; in history we find mathematicians, such as G. Bernoulli, who were for and others, such as Gauss, who were against, and others, such as Leibniz, who remained undecided, finally others, such as G. Cantor, who were in favor of the actual infinite and against the actual infinitesimal considered as a segment of the rectilinear continuum. There was a lull, as it were, in the debate when analysis was given, with the help of the concept of limit, a firm basis in the field of finite magnitudes and when the tendency against the actual infinite and infinitesimal prevailed, as a result of an aborted attempt at a geometry of the infinite by Fontenelle.6 Nevertheless, the old debates resuscitated here and there, even though Gauss protested against the use in mathematics of a determined infinite magnitude. [198] It is a fact, however, that no one had ever defined correctly what they meant by the 'actual infinite' and 'actual infinitesimal'; which may have, as we have seen since, different forms; neither Bernoulli, nor the more recent idealist of du Bois-Reymond7 had defined them. The definition of the actual infinitesimal by Poisson is not acceptable either. But, suddenly, light started to appear with the legitimate introduction 169 P. Ehrlich (ed.), Real Numbers, Generalizationsof the Reals, and Theories of Continua, 169-187.

© 1994 Kluwer Academic Publishers. Printed in the Netherlands.

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of actual infinite and infinitesimal magnitudes - with the transfinite numbers pf G. Cantor, the moments of Stolz 8 and the orders of infinity of functions of du Bois-Reymond. 9 But these did not deal with the actual infinite and infinitesimal of geometry. Cantor," using his transfinite numbers, claimed to have proved the impossibility of the actual infinitesimal as a segment of the rectilinear continuum. 0. Stolz" had already pointed out that the problem of the existence of the actual infinite and infinitesimal depends on an axiom, according to which, given two rectilinear segments, one smaller than the other, there is always a multiple of the former larger than the latter. Stolz gave to this axiom the name of Archimedes because it is axiom V of the work The Sphere and the Cylinder of the great Syracusan; but it had already been used by others.' 2 Stolz"3 has pointed out that Cantor's proof could affect neither his moments, nor du Bois-Reymond's orders of infinity, which, while not satisfying the Archimedean axiom, are not linear magnitudes;"4 but Stolz had also maintained that an infinitesimally small rectilinear segment is impossible, by giving a proof of this axiom based on the postulate of the continuum in the form given by Dedekind.' 5 We can also deduce the Archimedean axiom from the forms of the postulate of the continuum given by Weierstrass and Cantor, which are more appropriate to the calculus.16 Therefore, it is not a question of whether there exists actually infinite or infinitesimal magnitudes, but rather if there exists rectilinear segments satisfying the fundamental properties of the straight line, with the exception of the Archimedean axiom. The usual paths seemed closed after the proofs of Cantor and Stolz. It was therefore not through the path of analysis that these segments could have spontaneously presented themselves, since the authors just mentioned were starting from the one-to-one correspondence between the intuitive and the numerical continuum or from the transfinite numbers of Cantor, which seemed to be the only transfinite numbers. Similarly it is not through analysis that my general space of an infinite number of dimensions' 7 could have spontaneously presented itself, since we could consider only varieties with a finite number of variables. [199] Therefore, the answer had to be given through the rectilinear continuum itself, considered intuitively and decomposed in its possible elements, and we then realised that the postulates of the continuum mentioned above contain something which is not necessarily suggested by the continuum itself. In fact, this continuum is given to us by

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experience; it is an arbitrary operation to fix some points for its determination or for the practical operations that we must do with it. If we idealise the point by regarding it as the extremity of a line, we can see that it still cannot serve to form the continuum, because we always find ourselves in the presence of a segment which contains, ideally, many other points distinct from the extremities. The postulate according to which a point corresponds to every rational number is not verified in practice and in idealising the point and the segment in such a way that the latter always contains points distinct from the extremities, the one-to-one correspondence between the points of the straight line and the ordinary real numbers is no more justified. I had already pointed out to Stolz that the Archimedean axiom is derivable from Dedekind's postulate of the continuum because the latter is based on the same correspondence, while we can separate the Archimedean axiom from that of the continuum by giving to the latter the following form: If in a segment AB there exists a variable segment XX' such that AX is always growing and smaller than AX' which is always decreasing, and XX' becomes indefinitely small (i.e. smaller than any given segment), then it contains a point Y distinct from X and X'. 18 We add to the postulate of the continuum in this new form a new one, analogous to that of Archimedes, i.e. that if a and 0 are two rectilinearsegments, such that a is smaller than 0, we can construct a multiple of oa (according to a symbol of multiplicity TI) which is greater than P. Evidently if il is a finite natural number, this postulate becomes that of Archimedes. In my Fondamenti, I have precisely constructed actually infinite and infinitesimal segments which satisfy the condition that, a being given as the unit, we can construct 0 and vice versa."9 With these segments, we can do all the ordinary operations of addition and substraction; we can find the multiples and submultiples of these segments, and finally carry out the rational and irrational operations on these, in such a way that we can carry out the fundamental operations governed by the ordinary rules, with the symbols (numbers) representingthese segments. The question of the existence of these actual infinite and infinitesimal segments being raised first, as it should have been, the arithmetic conception of these numbers had to stay in the background, because, as I said, it was advantageous to tackle this question from the angle of geometry, rather than from the angle of arithmetic. Though the insufficiency of the arithmetical development was under-

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standable, it nevertheless led to some of the criticisms directed against these new infinities and infinitesimals. This is why Levi-Civita, [200] when he was still a student and at my suggestion, was the first to deal with this question from the angle of arithmetic; he did this by constructing the above mentioned numbers arithmetically and by completing them by introducing new units which were necessary for other operations.20 From another direction, Hilbert, 2' by constructing a non-Archimedean geometrical field, confirmed with his authority the logical possibility of such a geometry; Bindoni,2 in his doctoral thesis, proved that Hilbert's geometrical field was included in mine. Recent research in set theory, particularly that of Schoenflies, 2 3 confirms the logical validity of this geometry. The most recent research on the problem of the rectilinear continuum acquires particular interest, because it remains to be shown definitively that, as I believe, there is only one type of non-Archimedean number which satisfies it, adding other units, if need be, to complete it. 26 25 24 These are questions with which Hahn, Schoenflies and Vahlen recently occupied themselves. The logical validity of the non-Archimedean rectilinear continuum being thus established, then by the same token so is that of nonArchimedean geometry, for which I have chosen in my Fondamenti the Riemannian form: in this geometry, in an infinitesimalfield surrounding a point, if we consider only segments which are finite relative to one anotheror which mutually satisfy the Archimedean axiom, then Euclidean 28 geometry holds.2 7 This theorem has since been proved by Levi-Civita for the non-Archimedean geometry of Euclid and Bolyai-Lobatchevsky. We can consider as corollaries of this theorem those of Dehn,2 9 which were found by following the methods of Hilbert, on the relations between the sum of the angles of a triangle and the parallels carried from a point to a line; which means that there exist two non-Archimedean geometrical systems in which the sum of the angles of a triangle is greater than or equal to two straight angles and in which many parallels to a given line pass through a point." The logical validity of non-Archimedean geometry brings with it the independence of the theories of proportions and projectivity from the Archimedean postulate; theories of which other geometers took care of by simpler methods, in particular Hilbert and Schur.31 *

*

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Once the validity of non-Archimedean geometry is established, there remains the question of its content and method, which were objects of criticism, although they were less precise. Allow me gentlemen, as much as time allows me, to address this point. To someone who has the habit, in the superior researches of science, of only taking care of results and does not give importance to the content of mathematical objects and to method, these issues may appear to leave the field of mathematics; but, on the contrary, the content is by itself an essential element of science, and if the method is not carefully chosen it can also lead to petitiones principii. Here I shall use in another form considerations, already old, that I have developed in my Fondamenti di Geometria and earlier [201] in lectures given at the University of Padova between 1885 and 1890, which served as the basis for the publication of the Fondamenti itself. I shall also take into account later publications. The objects of pure mathematics do not necessarily have a representationoutside of thought; number, for example, is in its first formation the result of the mental operation of enumerating objects which are also abstract. Truth has its first foundation in the logical principles and in simple mental operations which are universally agreed upon. The liberty of the mind in its creations is limited only by the principle of contradiction, whence it follows that an hypothesis is mathematically possible when it is not in contradiction with the premises. According to us, pure mathematics, like formal logic, is exact. Geometry, on the contrary, has its origin in the direct observation of the objects of the external world, which is the physical space, and from the idealized observation of these it extracts its first and precise undemonstrable truths, necessaryfor its theoretical development. These are, in the proper sense of the word, the axioms. For example, the axiom in virtue of which between two points of the field of our observation only one rectilinear object passes. But in order for geometry to be exact, it must represent the objects given by observation by means of abstract or mentalforms and the axioms by means of well-determined hypotheses, independent of spatial intuition, in such a way that geometry becomes a part of pure mathematics or of the theory of abstract extension (Ausdehnungslehre). Here the geometer proceeds with his constructions without the need to see if they have an external representation or not, until he applies them to the physical world, without thereby abandoning the vision of the figure and all the advantages derived from the use of

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intuition in geometrical research. Yet, the exactness of geometry will be all the more great because of the certainty of the axioms suggested by observation and, consequently, because of their simplicity and small number.32 Indeed, observation is only vague and sometimes also deceptive and illusory; for instance, when we see the size of objects changing while we are moving, though the laws of perspective tell us that there is no such thing. Admittedly, the requirement of simplicity and of the least number of axioms leads to unavoidable and meticulous research. This meticulousness causes one to lose sight of the general concepts and constitutes one of the difficulties in reading the results of this research, when we suppose that nothing is mathematically written down yet and consider on the whole [202] the problem of the principles as in my Fondamenti. It is also clear that the axioms must be agreed upon universally, and for this we can admit as evident, and therefore undemonstrable, only the axioms accepted by the empiricist philosopher,for whom it is useless to give a proof of their logical compatibility. But such a proof is on the contrary necessary when we extend these axioms to an unlimited space, because no one has ever or will ever observe such a space. This is why we cannot accept as an axiom suggested by observation that of the parallels, when we define these straight lines as lines of the plane which do not meet when indefinitely extended, because no one has ever observed two such straight lines. Similarly, for example, we cannot admit as a primitive axiom drawn from observation that according to which the unlimited straight line is an open linear system. But the axioms drawn from pure observation are not sufficient for geometrical research. Geometry having become a part of pure mathematics, or rather of the theory of abstract extension, we now admit in geometry all the hypotheses or postulates which do not contradict each other or other already admitted axioms; these hypotheses either limit or extend the field of geometry, for example the postulates of Archimedes, of the continuum and of spaces with more than three dimensions, et cetera, or they serve to choose one of the possible forms determined by already given axioms or hypotheses, such as the postulate of the parallels.33 From these remarks it also follows that one must distinguish physical space from intuitive space, which is both an idealised representation of the former and an intuition; and intuitive space from abstract geometrical space, which is a concept. These forms were not properly distinguished, even by eminent authors such as Helmholtz. Abstract

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geometrical space is precisely the part of pure extension in which intuitive space is represented, but conversely there is no effective or even approximate representationof all its forms, and it is not necessary that there be such a representation in physical or intuitive space. So much so that not only is the equality of geometrical figures not necessarily determined by the movement of rigid bodies, as Helmholtz believed, but it is rather the equality of geometrical figures (which depends in turn on the logical concept of the equality of two different objects) which is necessary to define the movement of rigid bodies. From this also follows another consequence: that theoretical geometry is neither a part of mechanics, as Newton believed, nor does it depend upon physics, as Helmholtz thought. The distinction between physical space and geometrical space entails postulates which are necessary only for the practical applications of geometry, such as that of the approximate movement of rigid bodies, that of the three dimensions and that of Archimedes, whereas there are postulates of geometrical space, such as that of the general space and that of the non-Archimedean continuum, which we do not need to admitfor physical space.34 [203] Since intuitive space is represented in the geometrical space (with infinitely many dimensions, as I defined it in my Fondamenti) we can work with intuition, imagining in it the point, the straight line and the plane as in ordinary space and act as if in pure geometry. But, neither having nor being able to have a natural intuition of fourdimensional space, we combine intuition with abstraction, as we do in order to obtain the unlimited space from the intuitive one, and the habit we thus acquire is such that, as we believe that we have an intuition of the whole unlimited space, we believe that we see in a four-dimensional plane two planes meeting in a single point.35 In the distinction between physical space and geometrical space, Mill's assertion that the mathematical straight line does not exist in nature (or, what is more appropriate, in physical space) is reconciled with Cayley's remark that we couldn't assert this if we didn't have the concept of a straight line. Therefore, in geometry the liberty of the mind is not only limited by the principle of contradiction, but also by the data of spatial intuition. If it was proven that Euclid's postulate was intuitively valid (as the Kantians hold), we would not admit, for example, a plane in which Desargues' theorem on homologous triangles does not hold (Hilbert), or a plane in which a straight line rotating around a point cannot assume

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the position of another straight line passing through the same point (Poincar6Q, or the planes of Bolyai-Lobachevsky, Riemann or elliptic geometry. Similarly, if Euclid's postulate was physically or intuitively valid, we could not admit a geometry in which the straight line was determined by three instead of two points, while these forms are possible in abstract extension and can have, in their entirety or in part, a representation in geometry itself in the same way that the two-dimensional varieties of Riemann, Bolyai-Lobachevsky and elliptic have representations, respectively, in the geometry of a spherical surface, the pseudosphere and the infinite improper plane. There is here a contrast, but not a contradiction, with the principle according to which for certain categories of properties we can consider two different objects as equivalent, for example two forms [204] which can be transformed in each other projectively or birationally, because with this principle we do not take into account the other geometrical properties and the content of these objects which constitute, on the contrary, their essence. For example, physical space and geometrical space are, in virtue of their content, substantially different from each other, as they are different from the analytic varieties which represent them. And just as existence constitutes an essential element of physical space, so construction (of the geometrical space) constitutes an essential element of geometry, which should not be forgotten, as usually happens. That this content has such a fundamental importance is shown by the fact that Cayley, who initiated the use of the projective method in non-Euclidean geometry, considered Euclidean geometry to be valid in an absolute sense; this is why in Cayley's research it is more a question of a Euclidean representation of non-Euclidean geometry, which is brought about by modifying the notion of distance, in the same way the pseudosphere, the sphere and the infinite improper plane are representations of non-Euclidean geometry in the Euclidean one. Now, on the contrary, the content of these geometries has a remarkable importance: it tells us that actual external observation is not sufficient to establish one or the other geometry. As we can see, this content has a philosophical import on the form of the space, while Cayley's research could not have any, as the theory of imaginaries or that of the improper infinite didn't have, since they deal only with names referring to already existing and effective objects which add nothing to the origin of the space. From all this it also follows that mathematical research on the principles of science is, and must be, held clearly distinguishedfrom

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philosophical researchon the origin of mathematical ideas; and I, myself, when determining the content of the objects of pure mathematics and of geometry, didn't attempt to take sides for one or another philosophical system, since when I said that numbers do not necessarily have a representation outside thought, I did not want to say that they are not of empirical origin; equally when I said that the point has an empirical representation, I did not want to say that it is not a pure a priori intuition of the mind necessary to all external experience. This distinction is fortunate, because mathematics unites us while philosophy, at least until now, divides us. It is true that the study of the principles of science has provoked and will continue to provoke discussions between mathematicians, but an error in mathematics will always tend to be eliminated, while new ideas which are definitely acquired by science will remain. Errors depend either directly on the mathematician or on the vagueness of new ideas or on the lack of clarity in which they are first presented; but errors also come from the difficulties which these ideas face at the beginning when they run counter to deep-rooted convictions which are reinforced by the authority of eminent mathematicians, or counter to the lack of concern of those who, wanting to avoid having to make the effort of thinking, would exclude from mathematics research on the principles of science, or counter to the opposition of others to whom the new thinkers are the revolutionaries of science. And to the darkening of the nascent light of new mathematical truths, contributed those philosophers who, confident in already known mathematical principles, saw or thought they saw in new ideas an offence against their hypotheses concerning knowledge and the interpretation [205] of nature, while a new arrangement of the principles suggested and reinforced by new facts may not only be profitable to mathematics, but also to philosophy. Philosophy must accept new mathematical ideas when they are definitively formed. If, however, mathematical research must be distinguished from philosophical research, it would be advisable that the mathematician abstains from justifying his concepts by philosophical considerations or by fictions which easily lend themselves to philosophical criticism, like du Bois-Reymond does with his empiricist and idealist or like Cantor does sometimes to justify his transfinite numbers, which have a legitimate existence, despite some recent philosophical criticisms. But, on the other hand, the mathematician must not, in fear of such criticism, take refuge in a purely abstractfield or in a symbolic formalism, showing indifference to the questions of the content of

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mathematics, as has happened in the past and still happens now, when geometry is confused with the general theory of varieties of purely abstract elements. It is thus preferable that the ordering of the principles follows the logical and simplest development of mathematical ideas and thus that the method should neither be a lifeless device nor appear as a mere game of symbols or words, however useful it may be, but should be philosophical.Thus mathematics may also be useful to philosophical research on the origin of mathematical ideas, in the same way that it has the task of being useful to the applied sciences, which have as their direct objective the study of natural phenomena, by choosing the approximate methods most appropriate to this goal. And when one follows on the contrary an indirect method, for example, representing space by a variety with multiple variables in order to study its principles, it is necessary to examine if, while following the content of the space or of its construction, the postulates of the said variety can be justified without recourse to the concepts which are defined by these postulates, because this recourse would be a petitio principi and a philosophical mistake. On the choice of method, the most eminent mathematicians agree. Du Bois-Reymond remarked that if in operating with the signs of pure mathematics we forget about their meaning, we must not forget their origin in the discussion of the fundamental concepts of mathematics; and for geometry, Newton rightly noticed that the simplicity of the figure depends on the simplicity of the origin of the ideas, that is not of their equation but of their description; and Gauss maintained that for the connection and representation of geometrical truths, logical means cannot produce anything by themselves and come into bud without producing fruits, when fertile and vivifying intuition does not dominate everywhere. Weierstrass, Lie and Klein expressed similar views. My Fondamenti di Geometria conforms to these concepts, in particular in the sections dealing with the origins of non-Archimedean geometry. There remains, however, the fact that this method, without the contribution of analysis, when we suppose on the contrary that nothing is mathematically known, renders the reading difficult; and it is only in recent years that, in Italy and elsewhere, the method based on pure reasoning prevailed in the research on the principles of geometry. Indeed, everybody remembers the fate of Grassmann's Ausdehnungslehre of 1844, which is certainly preferable to that of 1862.36

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[206] To come back to non-Archimedean geometry, it is necessary to ascertain that it satisfies the aforementioned conditions. Examining the continuum as it is given to us by direct, raw observation, the Archimedean axiom is found to be valid for two rectilinear objects, because no matter what they are, even if we cannot construct in practice a multiple of one larger than the other, we could consider an nth part of these two objects sufficiently small such that a verification of the axiom is possible for these parts, and thus for the objects themselves. But the extension of this axiom to the whole unlimited space is not equally justifiable. Indeed, when we admit that in every idealised segment there are points distinct from the extremes, neither observation nor intuition can lead us to establish the Archimedean axiom between two segments which cannot be observed. And since we prove that there exists an actual infinitesimal segment which can be considered, with an infinite approximation, as nil with respect to a finite segment, we conclude that if such a segment was also to exist physically, we would not be able to see it. Nevertheless, we can use our intuition in any field of finite segments which satisfies the Archimedean Axiom. Thus, non-Archimedean geometry satisfies the conditions which are in general imposed on geometry by spatial intuition, and its content is thus geometrically justified.

But another problem, also geometrical, presents itself as a result of our premises: can unconfirmed hypotheses, thanks to further more accurate or more extended observations, be discovered to have an effective representation in the physical world? Among these hypotheses, the most characteristic ones are those of the parallels, the continuum and the hyperspace. We have already observed that if the Euclidean hypothesis was to be excluded, we would not be able to speak of a Euclidean space. For the rectilinear continuum, we observe on the contrary that the existence of the actual infinite and infinitesimal does not contradict our intuition; however, since no experience will ever lead us outside of finite magnitudes, we can only say, stating an already mentioned theorem: if physical space is actually infinite with respect to the field of our observations, then in finite physical space, which is also supposed unlimited, Euclidean geometry would be valid. I have fought elsewhere against the contrary physical hypothesis of a space with four or more dimensions, associating myself with

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Helmholtz.37 At any rate, no application drives us to this purely fanciful hypothesis. It is strange, however, that some ideas emerged from such erroneous intuitions. Indeed, the idea of a space with more than three dimensions was not born from Grassmann's Ausdehnungslehre, for which space and thereby geometry always have three dimensions; even less from the geometrical nominalism of Cayley, Cauchy, Riemann and others, found in their studies of certain analytic varieties with more than three dimensions, and no more than from my geometrical construction of hyperspaces. Rather, it sprung up from the physical hypothesis itself, which preceded the above and which has prevented the acceptance of the mathematical hypothesis and led the vulgate to confuse the defenders of geometries of more than three dimensions with spiritualistsand the 38 so-called mediums ei la Z611ner. [207] As for the usefulness of non-Archimedean geometry, I remark that we cannot confuse this geometry with just any other geometry which would have been obtained by neglecting or modifying a given axiom; for non-Archimedean geometry, like non-Euclidean geometry, has resolved a question which has been discussed for centuries and has shed new light on the formation of the continuum and geometrical space. This is sufficient for pure mathematics. Besides, every mathematical law being a law of thought, it is also a law of nature. Furthermore, because of the marvelous harmony existing between the laws of thought and those of the world, we cannot assert a priori that there could not be any useful application of the highest and most abstract mathematical conceptions. But this relative usefulness cannot be the direct goal of mathematical research in general and, in particular, of research on the principles of science. Nevertheless we do not exclude, on the contrary we insist, today more than ever, that one of the most important goals of science consists of satisfying the needs of the applied sciences and meeting as much as possible its important social function.

As geometers, we would stop here. But, although it is not our job to do philosophy, even though, as we have already said, we can admit as undemonstrable propositions only the minimum of simple facts to which the philosophers, that is the empiricist philosophers, have given their consent, we cannot accept the restrictions imposed by pure empiricism on the extent of mathematical research. Pasch himself, who made a useful

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attempt in this direction, which is laudable for other reasons, could not remain faithful to his program.39 Moreover, as geometers, we cannot accept, following the critique of Kant, that space and its postulates are a priori forms of pure intuition, because, for example, no mathematical proof was given of Euclid's parallel's postulate, which was the only one known to Kant. And the positivist philosophers fighting against the non-Euclidean hypotheses are no less metaphysicians than their Kantian colleagues. Even less can we accept that Euclid's postulate does not have the same evidence than the others and that it cannot be verified by subsequent observations, and yet that we have an a priorior subjective intuition of it such that subsequent observations cannot modify it, while admitting at the same time that this intuition derives from tactile and visual representations. 40 At any rate, we have no geometrical demonstration of the subjective necessity of Euclid's postulate and the other axioms, so that the geometer cannot accept the axioms given by the tactile and visual representations for the whole unlimited space without justifying such an extension. Our intuition is made of observations and of idealised experiences, because when I represent a straight line intuitively, I can imagine it only as an idealised rectilinear object, though afterwards I extend this representation by abstraction to any segment of the unlimited straight line. Indeed, we ascertain the presence of external objects and their properties by means of the senses and the quality of the sensations these objects produce in us, and in abstraction we keep only that of extension in order to obtain the first geometrical forms. In this way, spatial intuition is, as language is, the product of long experience. Men [208] possess it in varying degrees; it should be more perfect in geometers and painters, and insufficient in those who, blind from an early age, regain sight with an imperfect intuition of the simplest geometrical forms. B. Russell 4" raised the question of the a priori in a new way by distinguishing between the logical a prioriand the psychological a priori, which nevertheless merge. But, although we agree on some fundamental considerations, including the independence of geometry from physics, we cannot agree, for example, with his demonstration that space must have, as a form of externality, only a finite number of dimensions, while the general space has an infinite number of them, nor with his other demonstration whereby he tried to prove that all the axioms common to the three non-Euclidean geometries are necessary for all experience, while, according to him, the parallel postulates are empirical. It would

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be appropriate to examine the consequences of these hypotheses on non-Archimedean geometry. There are concepts which are not given directly by observation, such as the unlimited, upon which we made the proof by complete induction depend, or the equality of figures independent of the principle of the approximate movement of rigid bodies. It is not yet clear which part of these concepts pertains to thought and which to experience.42 But, the fact that in geometry we substitute precise forms such as the straight line for the vague data of experience does not mean, as Klein seems to be maintaining,43 that the former are necessary forms of all experience, because the non-Euclidean postulates can be replaced by precise mathematical forms without being considered as transcendental forms of our spirit. It is certain, however, that theoretical geometry has its origin in experience, but it renders itself independent from it by the exact formulation of its axioms and hypotheses by the proof of the possibility of these hypotheses and axioms when they are extended to the unlimited space and by constructing forms that are not suggested by experience. These forms are, however, constructions brought about by axioms drawn from experience and developed by logical thought without contravening the conditions set by spatial intuition. The thought, the psyche and the senses are so intimately connected that to separate what is specific to each is almost always an arduous, if not unsolvable, problem, so that philosophy has been circling for centuries without penetrating these problems and obtaining a complete solution. It is only with the specialisation of research and with an experimental and scientific orientation that one could arrive for some problems at, at least, a clear and sure philosophical synthesis, of which the specialists could draw up the elements. If within these problems we consider those concerning mathematical ideas, the contributions to their solution brought about by the mathematicians is one of the most beautiful monuments of science. Translatedby Mathieu Marion (with editorialnotes by Philip Ehrlich) NOTES Translator's note: This paper appeared originally in the Atti del IV Congresso Internazionale dei Matematici (Roma 6-11 Aprile 1908). (Rome, 1909), Vol. 1, pp. 197-208. The original pagination has been inserted in the text, in order to facilitate

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reference. A French translation was also published in the Bulletin des sciences mathdmatiques, 2i~me s6rie, 33 (1909), pp. 186-204. The following footnote was added to the translation: "Following the kind invitation of Mr. Darboux, I am pleased to publish in the Bulletin des sciences mathdmatiques a French translation of my lecture, which has been published in the Atti del IV Congresso Internazionaledei Matematici (Roma 6-11 Aprile 1908)". There is no indication as to who translated the text, but the translation contains many noticeable improvements and the suggestion seems to be, therefore, that Veronese translated his own text into French, although we do not know this for certain. Where the two texts differ, we usually followed the French version. The translator would like to thank Verity Johnson, Giuseppe Varnier and especially Philip Ehrlich for their help. 2 Editor's note: Veronese's mathematical and philosophical writings are notorious for containing poor grammar and tortured sentence construction. For the sake of readibility, an effort has been made to repair the most blatant examples of these types of defects. To distinguish Veronese's notes from those of the editor, henceforth editorial notes will be indicated by square brackets. 3 [3rd International Congress of Mathematicians, Heildelberg 1904.] 4 [Poincard, H.: 1903, 'Review of Hilbert's Foundations of Geometry', Bulletin of the American MathematicalSociety, series 2, 10, 1-23. (This is E. V. Huntington's translation of the original French version in Bulletin des sciences math~matiques, 2i~me sdrie, 26 (1902), pp. 240-272.) See also 'Extrait d'une lettre de M. H. Poincar6', Bulletin des sciences mathimatiques, 2i~me s~rie, 27 (1903), p. 115, and 'Rapport sur les travaux de M. Hilbert', Bulletin de la SociNt• Physico-Mathimatiquede Kazan, S6rie 2, 14 (1905), pp. 10-48.] 5 The lecture did not take place in the Congress, because the author became ill as soon as he arrived in Rome, and he did not achieve one of his aims, which was to promote a debate within the Congress on these questions. 6 Elements de gdomrtrie de l'infini (Paris, 1727). See my Fondamenti di Geometria, 1891, p. 620; German translation by A. Schepp [GrundzUge der Geometrie], 1894, p. 697. Contrary to the claim of Cantor (Mathematische Annalen, 46), the infinities of Fontenelle have nothing to do with non-Archimedean geometry. 7 [See Chapter 1 of Paul du Bois-Reymond's Die Allgemeine Funktionentheorie, TUbingen, 1882.] S[Stolz, 0.: 1884, 'Die unendlich kleine Grdssen', Berichte des NaturwissenschaftlichMedizinischen Vereines in Innsbruck, 14, 21-43; and Vorlesungen iiber allgemeine Arithmetik (ErsterTeil: Allgemeines und Arithmetik der reelen Zahlen), Teubner, Leipzig, 1885, pp. 205-216.] 9 [See, for example, P. du Bois-Reymond, 'Sur la grandeur relative des infinis des fonctions', Annali di Matematica Pura e Applicata, serie Ila, 4 (1870-71), pp. 338-353; 'Ober die Paradoxen des Infinitarcalciils', MathematischeAnnalen, 11 (1877), pp. 149-167; and the reference in note 7.] 1o [Cantor, G.: 1887, 'Mitteilungen zur Lehre vom Transfiniten', Zeitschrift fur Philosophie und philosophische Kritik, 91, pp. 81-225, 92, pp. 242-265.] "H [Stolz, 0.: 1883, 'Zur Geometrie der Alten, insbesondere Oiber ein Axiom des Archimedes', Mathematische Annalen, 22, 504-519. This is a revised version of a paper of the same title published in Berichte des Naturwissenschaftlich-MedizinischenVereines in Innsbruck, 12 (1882), pp. 74-89. Stolz first isolated and stated the importance of the

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Archimedean axiom in the appendix to his historically important paper which brought Bolzano's contributions to the calculus to the attention of XIXth-century mathematicians, 'B. Bolzano's Bedeutung in der Geschichte der Infinitesimalrechnung', MathematischeAnnalen, 18 (1881), pp. 255-279.) 1Z See my Fondamenti di Geometria, Appendix. On recent discussions of Cantor's tranfinite numbers, see Shoenflies, Die Entwicklung der Lehre von den Punktmannigfaltigkeiten, 1908. On the infinities of du Bois-Reymond, see the recent writings of E. Borel and E. Bortolotti. 13 [Stolz, 0.: 1888, 'Ober zwei arten von unendlich kleinen und von unendlich Grissen', Mathematische Annalen, 31, 601-604.] 14 [In the paper cited in note 10, Cantor introduced a concept of a linear magnitude in the hope of being able to prove the impossibility of infinitesimal line segments, by showing that this concept, in conjunction with his theory of transfinite ordinals, implies the Archimedean condition. Cantor concluded from this that the so-called Archimedean axiom is not an axiom at all, but rather a condition which followed with necessity from the concept of linear magnitude.] "15[See the paper of 1883 listed in note 11. It is worth noting that while Stolz is generally credited with having first proved that Dedekind continuous ordered Abelian groups are Archimedean, Veronese (see the reference in note 19) pointed out that Stolz's original 'proof is flawed. Stolz later acknowledged the point and sought to rectify the problem in his paper 'Ober das Axiom des Archimedes' (Mathematische Annalen, 39 (1891), pp. 107-112). Before the paper went to press, however, Stolz added a footnote (p. 108) pointing out that he had recently learned that Bettazzi had already published a proof of the theorem in his monograph Teoria delle grandezze (Pisa, 1890). The most elegant of the early proofs of this theorem, however, is the famous one given in 1901 by Holder in the paper cited in note 18.] "16Professor Enriques, who referred precisely to non-Archimedean geometry in his work on the principles of geometry (Encyklopadie der Mathematischen Wissenschaften, III, 1907), nevertheless makes a mistake when he gives to the postulate of Cantor (Mathematische Annalen, 5), a form equivalent to mine and concludes that we cannot deduce the Archimedean exiom from Cantor's postulate. "7 [Veronese was one of the earliest contributors to the study of n-dimensional geometrical spaces. In his Fondamenti di geometria (1891), he also introduced and studied in great detail the idea of an infinite dimensional space. For a discussion of his conception of a general space of an infinite number of dimensions, see Chapter One of Book One of his Fondamenti di Geometria.] 18 [This intended generalization of the Dedekind continuity condition first appeared, in a slightly different form, on page 612 of Veronese's paper (correctly cited by the editor in Veronese's note 19). (A brief discussion of the first formulation can be found in the editor's introduction to this volume.) Otto Holder later provided the following crisp formulation of the condition for the strictly positive cone of a totally ordered group S: If X, Y are non-empty subsets of S where X has no greatest member, Y has no least member, and every member of X precedes every member of Y, then if for each (necessarily positive) member z of S there are elements a of X and b of Y for which b - a < z, then there is a c in S lying between the elements of X and those of Y. See page 10 of Holder's paper, 'Der Quantitat und die Lehre vom Mass', Berichte uiber die Verhandlungen der Koniglich Sitchsischen Gesellschaft der Wissenschaften zu Leipzig,

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Mathematisch-Physische Classe, 1901, pp. 1-64. Despite the fact that Veronese's continuity condition was widely discussed during the early part of the 20th century, his name is rarely associated with the condition today. Holder's formulation of the condition (or some simple variation thereof) has been rediscovered numerous times throughout the 20th century and today the condition is regarded to be of considerable importance in the theory of ordered algebraic systems. Its full significance began to be appreciated when Cohen and Goffman proved that every ordered Abelian group has a so-called Dedekindian (i.e. Veronesean) completion which is itself an ordered Abelian group. (See 'The Topology of Ordered Abelian Groups', Transactions of the American Mathematical Society, 67 (1949), pp. 310-319). Banaschewski extended the result to arbitrary ordered groups. (See, 'Uber die Verollstindigung geordneter Gruppen', Mathematische Nachtrichten, 16 (1957), pp. 51-71.) Several mathematicians later showed that a similar result holds for ordered fields. (See, for example, D. Scott, 'On Completing Ordered Fields', in Applications of Model Theory to Algebra, Analysis and Probability Theory (InternationalSymposium, Pasadena California, 1967) W. A. J. Luxemburg (ed.), Holt, Rinehart & Winston, 1969.)] "19See also my 'I1 continuo rettilineo e l'assioma d'Archimede' (Atti della Reale Academia dei Lincei, 1890), and Holder, Der Quantitdt und die Lehre vom Mass (LeipzigBerlin, 1901). In non-Archimedean geometry it is also possible to speak of the measurement of segments when one of them is taken as the fundamental unit of measure. [Editor's note: the citation provided by Veronese of his own paper is incorrect. It should be: Atti della Reale Academia dei Lincei (Roma), Memorie della Classe di Scienze Fisiche, Matematiche e Naturali, serie 4, 6 (1889), pp. 603-624.] 20 [Levi-Civita, T.: 1892-1893, 'Sugli infiniti ed infinitesimi attuali quali elementi analitici', Atti del Reale Instituto Veneto di Scienze Lettere ed Arti, Venezia, (7), 4, 1765-1815. The 'units' which Veronese refers to are the 'component terms' (with coefficients deleted) in a formal power series. Levi-Civita generalized the construction of a non-Archimedean ordered field given in the above paper in 'Sui numeri transfiniti' (Atti della Reale Academia dei Lincei, classe di scienze fisiche, matematiche e naturali, Rendiconti, serie Va, 7 (1898), pp. 91-96 & 113-121).] 21 [Hilbert, D.: 1899, Grundlagender Geometrie, in Festschriftzur Feier der Enthillung des Gauss-Weber-Denkmals, Leipzig, Teubner.] 22 [Bindoni, A.: 1902, 'Sui numeri infiniti ed infinitesimi attuali', Atti della Reale Academia dei Lincei, classe de scienze fisiche, matematiche e naturali,Rendiconti, 9, 205-209.] "23[Schoenflies, A.: 1908, Die Entwicklung der Lehre von den Punktmannigfaltigkeiten, Bericht erstattet der deutschen Mathematiker-Vereinigung,zweiter teil, Leipzig, Teubner.] 24 [Hahn, H.: 1907, 'Uber die nichtarchimedischen Grossensysteme', Siztungsberichte der kaiserlichen Akademie der Wissenschaften. Mathematisch-Naturwissenschaftliche Klasse, 116, section Ila, 601-655.] "25[Schoenflies, A.: 1906, 'Uber die MOglichkeit einer projektiven Geometrie bei transfiniter (nicht-archimedischer) Massbestimmung", Jahresbericht der deutschen Mathematiker-Vereinigung, 15, 26-41.] 26 [Vahlen, T.: 1907, 'Uber nicht-archimedische Algebra', Jahresberichtder deuschen Mathematiker-Vereinigung, 16, 409-421; and Abstrakte Geometrie, Leipzig, Teubner, 1905.] 27 [Veronese is attempting to convey the following idea: let AB and YZ be segments

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of a non-Archimedean Riemannian space and further suppose that AB is infinitesimal relative to YZ, i.e. nAB is smaller than YZ for all positive integers n. Let G be the region of the space containing all and only those segments AC such that AC is finite relative to AB (i.e. for each such AC there are positive integers m and n such that nAC is longer than AB and mAB is longer than AC). G is an example of an infinitesimal field surrounding A (relative to YZ). Veronese is telling us that if we limit our attention to geometrical figures in G which are bounded by segments which are finite retative to AB, then the theorems of the geometry governing those figures may be interpreted with an infinite degree of approximation as identical with those of ordinary Euclidean geometry. I hasten to add, however, that the segment AB might be finite relative to ordinary objects such as the standard meter stick.] "28[See the first paper listed in note 20.] 29 [Dehn, M.: 1900, 'Die Legendreschen SAtze uber die Winkelsumme im Dreieck', Mathematische Annalen, 53, 404-439. An observation to the effect that Dehn's work on non-Legendrian geometry could be studied in connection with Veronese's work on non-Archimedean Riemannian geometry had already been made by Federigo Enriques in his famous Encyklopedia article 'Prinzipien der Geometrie' referred to in note 16. See Enriques' note 263 on p. 128.] 30 [Indeed it suffices that we consider an infinitesimal non-Archimedean field in which the absolute sum of the angles of a triangle is, in Riemannian or elliptic geometry, greater than two straight angles and, in Euclidean geometry, equal to two straight angles; and infinite number of parallels to a given line pass through a point in this field, when we consider parts of these lines which are included in this very field.] 31 [Schur, F.: 1899, 'Uber der Fundamentalsatz der projectiven Geometrie', Mathematische Annalen, 51, 401-409. For Hilbert see note 21.] "32Klein also points out that the data from any observation are only worthy within certain limits of exactitude and under particular conditions, while when we fix the axioms we can put propositions of absolute precision and generality in place of the data, having recourse to Mach's principle of economy of thought, and so maintains that the axioms must be simple and in the smallest number (see Gutachen zur Verth. des Lobatsch. Preises, November 1897, Kasan, or Mathematische Annalen, 50 (1898); Vorlesungen uber die Nicht-Euklidische Geometrie, I, 1893). "3 For example in Hilbert's Grundlagender Geometrie, the system of the axioms appears like an arbitrary system of abstract truths, rather than a system of truths given partly by experience and partly as truths necessary to the logical development of geometry. 3' Veronese, G. 1891, Fondamentidi Geometria. The exclusion of the movement of rigid bodies from the definition of the equality of figures was welcomed by Hilbert (1899) and others; it was also accepted, and this was more difficult, in treatises of elementary geometry, by myself ([Editor's note: Elementi di Geometria], first edition, 1897) and later by Ingrami and Enriques-Amaldi. Although this exclusion, some traces of which can also be found in Euclid's Elements, was much discussed, it was never effectively obtained. (See G. Veronese, Fondamentidi Geometria, appendix.) B. Russell and Poincar6 consider that the possibility of the movement of an invariable figure contains a vicious circle. Poincar6 holds that the possibility of this movement is not a self-evident truth, or at least not in the same way as Euclid's postulate is. Indeed the empirical verification of the postulate of parallels can be made to depend on that of the movement of an

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invariable figure. But, by the distinction which I am making between geometrical space and physical space and consequently between pure geometry, for which the above principle is not necessary, and its practical applications, I disagree with the eminent French mathematician when he maintains (without making the above distinction) that "while studying the definitions of geometry, we see that we are forced to admit, without demonstration, not only the possibility of this movement, but also some of its properties". This principle and its properties are, on the contrary, necessary only for the practical applications of geometry, as is the axiom of the three dimensions of physical space. 3' This explains why we use the word 'space' instead of the word 'variety', which has greater but totally generic and abstract meaning. 36 [Grassmann, H.: 1862, Die lineare Ausdehnungslehre, ein neuer Zweig der Mathematik, Leipzig, 1844; and Die Ausdehnungslehre vollsttindig und in strenger Form bearbeitet, Berlin.] 31 Veronese, G.: Il vero nella mathematica (Inaugural lecture at the University of Padova, November 1906). 38 [Z611ner was an astrophysicist who thought that he had a means of experimentally proving the existence of a fourth dimension. Felix Klein had proven a theorem which showed that a certain knot which couldn't be untied in three-dimensional space, could be in four-dimensional space. Z611ner asked the famous American spiritualist Slade to see if he could untie such a knot by taking advantage of a fourth dimension. According to Z611ner, Slade succeeded! It is perhaps worth noting that Z611ner's original interest in four-dimensional space was not motivated by spiritualism but rather by his attempt to use such a space to explain Newtonian gravitational action at a distance. See Z611ner's Prinzipien eine elektrodynamischen Theorie der Materie, Wilhelm Engelmann, Leipzig, 1876, pp. lxvii sq., and for Felix Klein's reaction, see The Development of Mathematics in the 19th Century. Volume IX, Lie Groups: History, Frontiers and Applications, translated by M. Ackerman, Math. Sci. Press. 1979, pp. 158-160.] 31 Veronese, G.: Fondamenti di Geometria, appendix. 40 Enriques, F.: 1903, 'Sulla spiegazione psicologica dei postulati della geometria', Rivistafilosofica de G. Cantoni, Padoa; Encyklopidie der Mathematischen Wissenschaften, loc. cit., Einleitung. "41Russell, B.: 1897, An Essay on the Foundationsof Geometry, Cambridge, Cambridge University Press. 42 Veronese, G.: 'I1 vero nella mathematica', loc. cit. 4' Klein, F.: Vorlesungen uiber die Nicht-Euklidische Geometrie.

PART IV

EXTENSION AND GENERALIZATIONS OF THE REALS: SOME 20-CENTURY DEVELOPMENTS

HOURYA SINACEUR

CALCULATION, ORDER AND CONTINUITY*

We owe the possibility of making a clear distinction between the three terms which form the title of this paper to a relatively recent mathematical theory, the real algebra of Artin and Schreier (1926). The term 'real algebra' refers to an algebraic theory of real numbers; that is to say, an algebraic theory of the conceptual instrument which, from the Greeks to Cantor (and still later), has been used to render the linear continuum numerical. Real algebra is an innovation which has provided the starting point for numerous current developments in algebraic geometry and model theory, as well as in algebra, which have in common the feature that they imply an algebraic treatment of the continuum. Without going into technical details, 1 I would like to make several remarks concerning the essential elements which, taken together, characterize this innovation. I shall begin by recalling how the introduction of real algebra achieved a long-standing objective which up until then had been considered as desirable but probably unattainable. I shall then identify the immediate ancestral lines whose convergence gave rise to this new approach to the continuum. Finally, I shall point out that, by providing an algebraic foundation for the theory of inequalities, real algebra complements Steinitz's algebraic foundation of the theory of equations. I.

AN ALGEBRAIC FOUNDATION OF REAL NUMBERS

Real algebra is a product of the grand conceptual reform of 'modern' mathematics which during the 1930's advanced the two distinct but strongly related viewpoints of abstract algebra and abstract topology. It was elaborated between 1924 and 1927 by Emil Artin, a well-known algebraist famous as the driving force behind the mathematical seminar at the University of Hamburg, and Otto Schreier, one of the founders of topological algebra and author of the concept of a 'continuous abstract group. 2 Both Artin and Schreier belonged to the school of David Hilbert, the great champion of abstract methods and finiteness theorems, 3 which constituted according to him the major 'hallmark' of modern mathematics. Real algebra is a good example of an abstract method: it gives 191 P. Ehrlich(ed.), Real Numbers, Generalizationsof the Reals, and Theories of Continua, 191-206. © 1994 Kluwer Academic Publishers.Printed in the Netherlands.

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an algebraic theory of the ordered field of real numbers; or rather, to be more precise, it defines a class of algebraic structures, that of real closed fields, of which the real numbers are a model (a concrete realization which satisfies the axioms of real closed fields). Artin and Schreier succeeded in 'characterizing' the field of real numbers as a real closed field, thereby illustrating the idea, dominant at that time, according to which abstract algebra is simultaneously the instrument of expression as well as generalization. We may note that the sense of this 'characterization' ('Kennzeichung') which is implicit in the work of Artin and Schreier represents an innovative extension of the term. It is a fact that the real numbers are not determined up to isomorphism, as should be the case in order to conform to a strict usage of the term 'characterization'. However, they are defined as an element of a class of fields, and this class is characterized by the set of elementary statements4 satisfied by any of its members; all the fields in this class are thus equivalent with respect to elementary formulas which can be constructed in the language for real closed fields. Although the real numbers, R, and the real algebraic numbers, RA, are not isomorphic as Cantor showed in 1873, they are nevertheless elementarily equivalent (each being determined up to elementary equivalence). It is quite remarkable that the same elementary statements should be satisfied in a countable model, RA, and in a 'continuous' model, R. Cardinality is in a way 'neutralized' in first-order logic. The Ldwenheim-Skolem theorem, which at that time was not very well known to mathematicians who were not logicians, shows that this holds in full generality. Of course, second-order properties such as continuity, are also 'neutralized' in first-order logic. A real closed field is a commutative field K which satisfies (for example) the following three axioms: (1) the element -1 is not a finite sum of squares of elements of K; (2) every positive element of K possesses a square-root; (3) every polynomial of odd degree on K has at least one root in K. Artin and Schreier showed that the first axiom is equivalent to the statement: 'There exists a total ordering of K compatible with the field operations'. Thus, the first axiom does not express the fact that K is ordered, but only that K is orderable; it is this that the authors called a 'real field'. Thus, they made a remarkable semantic transformation in the adjective 'real', which traditionally applied to roots of polynomials (those which are not 'imaginary'), or to 'quantities', and then much later to

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numbers.5 Thus, a real field is a field admitting at least one, but possibly many, total orderings compatible with its algebraic structure. This is contrary to what we are used to with the ordered field of real numbers; we shall presently see why Artin and Schreier did not begin by considering ordered fields. However, they rapidly established ([3], Satz 1) that a real closed field can be ordered in a unique manner: it follows that there is no essential mathematical difference between 'orderable' and '(actually) ordered' when one is dealing with real closed fields. The conjunction of axioms (2) and (3) is equivalent to a maximality property. R, for example, is maximal in the class of commutative number fields which satisfy axiom (1); that is to say, real number fields (the adjective 'real' being taken here in its usual meaning). In other words, there exists no proper algebraic extension of R which can be ordered so as to extend the natural order of R. The same holds for the field RA of real algebraic numbers which is a subfield of R. On the contrary, if we start from the ordered field Q of rational numbers, it is possible to extend it algebraically while preserving its order, for example by taking the simple algebraic extension Q ('2). - We may note, in passing, that a real closed field is a real field which has no proper extension which is algebraic and real, but it is not necessarily an algebraic extension of its prime field: unlike RA, R is not algebraic over 0. Let us take a closer look at the formulation of the three axioms above. It is clear that what is at stake here is the provision of an algebraic expression for the existence (or, in the terminology of Artin and Schreier, of the 'possibility') of the natural ordering on R which is known to be a continuous ordering. We have indeed an expression in terms of squares, sums of squares, and polynomials and their zeros; i.e., in the last analysis, an expression in terms of just two operations, addition and multiplication, and a single relation, that of equality. To put things rather brutally in order to bring out the significance of the situation, I would say that it illustrates Hilbert's 'finitist' style, if not quite exactly then at least to a good approximation. Artin and Schreier's axiomatic system provides a countable algebraic description (i.e. a description formulated as a countable set of elementary statements exclusively constituted of algebraic symbols6 ) of a phenomenon, namely the continuum, whose essence is topological, and which from the point of view of the Cantorian theory of cardinality, is not only infinite but, moreover, uncountable. In short, an axiomatic system which captures the linear continuum by a countable set of algebraic statements. It

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is understandable that such an undertaking had been judged almost unthinkable before it was carried out, and paradoxical or amazing afterwards. The greatest mathematicians did not take exception on this score. Helmut Hasse witnessed at the time to the general surprise of seeing "order reduced to calculation" ([9], p. 27), and Andr6 Weil later underlined the apparent paradox involved in a theory of real numbers having a place in abstract algebra ([27], p. 174). It was natural enough that abstract algebra should succeed in mastering phenomena related to the algebraic structure of R (e.g. Galois and Steinitz's theory of equation solvability); but it would seem that it ought to have balked at the topological properties of R, regrouped in the intuitive idea of 'continuity'. The latter idea is expressed by a number of different statements; the fact that some of these statements are formally equivalent to each other does not necessarily imply that they are semantically homogeneous. The 'continuity' of R is, in fact, represented by any of the following properties: (1) the Dedekind cut property (to which we shall return below): for each partition of the rational numbers into two non-empty classes A, and A2 such that every element of AI is less than every element of A 2, there exists one and only one element of R less or equal than every element of A2 and larger than or equal to every element of A1; (2) the Cauchy convergence criterion for sequences of real numbers; (3) the Bolzano theorem or analytic intermediate value theorem (I.V.T.), which asserts that any continuous function f: ]a, b[ C R --- R, which takes values of opposite signs for two values x and 3 E ]a, b[, vanishes necessarily for some 7 between (x and P; (4) the Cantor theorem or 'nested interval property': in any sequence of nested real intervals a. < a,,, < . . . < b,,, < b. there exists a point common to all the intervals; (5) The least upper bound (L.U.B.) - or greatest lower bound (G.L.B.) - property for every nonempty, order-bounded set of real numbers. Now, although it is generally recognized that 'continuity' is a topological concept, it is by no means obvious that properties 1-5 all belong to the language of topology. For example, both 1 and 5 hinge more on the fact that R is an ordered field than they do on the fact that R is a topological field in the strict sense of the term (i.e. a field whose addition, multiplication and their inverses are continuous in the sense of preserving

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neighbourhoods'). The situation was thus overdetermined, due to the fact that R is a field which possesses both an order structure and a topology derived from the notion of distance (i.e. a metric). With the benefit of hindsight it appears clear that the definition of distance involves only the algebraic structure and the order relation; in other words, that it is possible to endow R (or any ordered field) which a metric without invoking topology as such. However at the time, with real analysis in full flow (Lebesgue, Borel, La Vall6e Poussin, etc.), it seemed strange to use algebra to disentangle the situation. This recourse to algebra might even have seemed superfluous, because at the same time the idea of continuity was being elaborated in the domain of general topology: preservation of neighbourhoods by continuous functions from R into R, connectivity expressing the fact that R 'consists of one piece', local compactness expressing the fact that every point in R has a neighbourhood which is compact (i.e. on which any filter has a limit point, or any ultrafilter is convergent), and completion expressed by the convergence in R (considered as a metric space) of Cauchy sequences. Nevertheless, according to Andr6 Weil, through their axiomatic system Artin and Schreier "hit upon the true foundations" of the theory of real numbers and "restored it to its proper place". What exactly does that mean? Just this: reverse the situation that made the most elementary (the study of the real solutions of polynomials with real coefficients) depend on the less elementary (the study of the more general notion of a continuous real function of real variables). It is clear that the notion of a continuous function from R into R covers far larger, more varied and more complex category of mathematical objects than that of a polynomial with real coefficients. It should thus be possible to treat the latter notion specifically by demonstrating all the theorems concerning polynomials under the more restrictive hypotheses which are associated with them. In short, it should be possible to treat real algebra without having to treat real analysis beforehand. This is exactly what Artin and Schreier's theory succeeded in doing. In the 19th century, the analytical formulation and demonstration of certain propositions were usually attributed to the 'analysis of equations', as it was called at the time. The axiomatic system of real closed fields made it possible to formulate and to demonstrate these propositions within the limits of algebraic language. This is the case for the Bolzano, Rolle and Sturm theorems, for the mean value theorem, for the theorem showing that the sum of the absolute values of the coefficients of a

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polynomial x' = ax"•' + . . . + a, is an upper bound on the absolute values of its roots, and for the theorem which states that any rational function whose denominator is nonzero on an interval takes on its bounds on that interval. All these theorems are true for polynomials and rational functions on R, not only because the latter are continuous functions from R into R, but directly as a consequence of the axioms for real closed fields. For this reason Artin and Schreier called them the 'theorems of real algebra', thus creating a new discipline which screens the usual propositions of real analysis in order to locate and distinguish those which can be assigned or reassigned to real algebra. In general, every question involving real numbers which is solvable within the general framework of real closed fields is an algebraic question. This result settles an old question. Indeed, ever since its origins, the search for the real roots of algebraic equations has presented mathematicians the problem of autonomy with respect to the concepts and methods of analysis, or in the terminology of the 17th and 18th centuries, of geometry. Every since the origin of the theory of equations, real algebra has been trying to extricate itself from real geometry. The autonomy of these two domains was an early object of conjecture, by Michel Rolle for example, but it had never really been assured. In 1798, Louis-Joseph Lagrange thought that he could succeed in separating the two subjects. At the beginning of his Treatise on the Resolution of Numerical Equations of any Degree, he states that the first principle of the numerical resolution of algebraic equations is proposition ([15], Theorem 1) which we call 'the intermediate value theorem' (I.V.T.) or 'the change of sign theorem'. This theorem is an algebraic version of the analytical theorem subsequently proved by Bolzano in 1817; its formulation contains only polynomials and change of sign conditions with no appeal to the notions of function and continuous variation. Artin and Schreier's theory has confirmed Lagrange's hypothesis by reformulating it in the general framework of real closed fields: I.V.T. is equivalent to the conjunction of axioms (2) and (3); I.V.T. is therefore the cornerstone of the algebraic theory of R and real closed fields. On the other hand, the Bolzano theorem characterizes R to the exclusion of any other real or even real closed field, as Detlef Laugwitz [16] has shown. However, it would not be correct to reduce Artin and Schreier's theory to that of Lagrange. The latter posed the right question but was unable to answer it. It fact, Lagrange was clearly aware that he had failed to achieve his objective of proving the first principle of

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the theory of equations by purely algebraic means, independently of 'the theory of curved lines'. His first proof ([15], p. 19-20), which is 'direct' and purely algebraic, does indeed beg the question. Lagrange therefore resigned himself ([15], Note I) to reasoning in terms of 'variations by insensible degrees' of curved lines, in other words to a geometrical justification that he even goes so far as to illustrate by a kinematic model. Artin and Schreier showed that Lagrange's second proof is unnecessary! His first proof is in fact quite correct, as soon as one has available Steinitz's theory of algebraically closed fields, and the equivalence 'K is real closed ý=• K(i) is algebraically closed' which they proved in their paper ([3], Satz 4). II.

HISTORICAL POINTS OF REFERENCE

It would be out of place to recall here the whole history leading up to Artin and Schreier's theory. A large number of results were involved, not least among them Steinitz's algebraic theory of fields and Hilbert's seventeenth problem. Of course I shall only consider those results which bear a direct relationship to the problem of expressing the 'continuity' of R. The construction of the set R by Dedekind and Cantor, and the characterization of R as a maximal archimedian ordered commutative field by Hilbert, are so well known that it would be superfluous to recall them in details. At the beginning of this century, Dedekind and Hilbert were both held in fervent esteem; it is therefore not surprising that Artin and Schreier's theory amounts to a sort of synthesis between Dedekind's construction and Hilbert's axiomatization. Let me explain. It is undoubtedly Dedekind who, by his project of giving "a purely arithmetical and perfectly rigorous foundation to the principles of infinitesimal analysis", highlighted the fact that R is an ordered set. Dedekind wanted to find "a veritable definition of the nature of continuity" ([8], p. 316) by reducing continuity to the ordering of R. Succeess in this task would achieve a double goal. Firstly, it would express arithmetically a notion 'represented' in geometry and analysis. This was attractive since arithmetic was held to be precise, whereas the domain of geometry and analysis was held to be conceptually rather vague and less well founded; according to Dedekind, the latter's rigour is 'borrowed' rather than intrinsic. Secondly, at the same time it would demonstrate the autonomy of arithmetic; there would not be the slightest trace left of any need for representations ('Vorstellungen') 8 foreign to

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arithmetic itself. This autonomy includes the capacity to express most if not all significant mathematical facts by its own means. At the head of this list is the 'continuity' of IR, which is the basis for so many arguments. In fact, it is well known that Dedekind started from the ordered set of rational numbers; showed that it has 'gaps' in the sense that there are nonempty bounded subsets of rational numbers without a lower upper bound or without a greatest lower bound; filled these gaps by considering the ordered set of all cuts of Q, which satisfy the L.U.B. and G.L.B. property; and thus finally defined the set of real numbers as the set of these cuts of Q. Thus, by reducing the idea of continuity to that of order, Dedekind initiated the distinction between ordered sets and topological spaces and showed that the topological structure of IR is not preponderant in every respect. But all this remained rather abstract, due to the fact that Dedekind did not make the connexion between R's field structure and its order, between calculation and ordering. R is on one hand a field (i.e. a set closed under addition, multiplication and their inverses), on the other hand an ordered set. To be more precise, the rules governing the compatibility of the order relation with the field operations were not formulated, because he failed to extend rational addition and multiplication to the whole set of cuts of Q; it was thus not yet possible to speak of 'ordered field'. Dedekind himself was keenly aware of this inadequacy: it is only in 1872 that he published a discovery dating back to 1858 which on his own admission had born "so few fruits"! And as a matter of fact, the majority of authors who expound Dedekind construction of the real numbers render it operational by borrowing more or less openly from the Cantorian construction using Cauchy sequences that can be added and multiplied. In contrast to Dedekind's construction, Hilbert's axiomatization [11, 12] does define R as an ordered field, and the "ordinary rules of calculation", to use the terminology of the 1900 lecture [13, p. 300], manifestly comprise both the laws for the operations of addition and multiplication and those which determine the compatibility of the order relation with these operations. In other words, manipulating the order relation is a part of calculating. On the other hand, Hilbert did not stop there, and added a group of axioms, called the 'continuity axioms', which make it possible to characterize R, up to isomorphism, as a maximal Archimedean totally ordered commutative field. Hilbert's concern was not to absorb the 'continuity' of R in the order structure; it is rather to

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find out in which cases an axiom of continuity (for example the Archimedes axiom which M. Pasch had already explicitly drawn attention to), is indispensable for the proof of a theorem and in which cases it is superfluous; in short, to establish what logically depends on continuity and what does not. Since he showed in the Grundlagen [11] that certain geometrical problems (for example, proof of the Pappus theorem) can be solved without recourse to the Archimedes axiom, this was not far from the analogous arithmeticalquestion of which problems involving read numbers can be solved independently of a continuity axiom. It is true that Hilbert did not pose this question explicitly. With their real algebra Artin and Schreier answered it: (1) by doing for R as a field what Dedekind had done for R as a set, i.e. by substituting a consideration of order for that of continuity in a whole category of problems; and (2) by showing, as Hilbert had implicitly suggested, that there is less difference between ordering and calculation than between the two taken together on one hand, and topology on the other. III.

THE ALGEBRAIC THEORY OF POLYNOMIAL INEQUALITIES

Today it is usual to work with fields which are ordered and real closed, 'maximal ordered fields' as Bourbaki says, because one then has an elementary theory with several 'good' logical properties which simplify the solution of many problems: elimination of quantifiers, logical completeness, model-completeness (i.e. the fact that every inclusion of models of the theory is an elementary inclusion), and so on. For example, the theory of R in the language L = {O, 1, +, ., =, >} has the three properties mentioned above. Now, we have pointed out that by virtue of the axioms defining it, a real closed field is only orderable and not equipped with an order a priori. There is all the more cause to ponder over the reasons that led Artin and Schreier to their choice, in that it not only does not correspond to our current practice, but does not follow tradition. For Hilbert, as for Dedekind, the order relation or R was primitive. At the beginning of their first paper, Artin and Schreier acknowledged that "it would be natural to start from the concept of an ordered field ".. . ". However, they prefered "a definition which employs only the operations of addition and multiplication, and which has the possibility of ordering the field as a consequence" ([3], in [2], p. 258). The situation is perfectly clear: in order to build an algebraic theory of R

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it must be possible to logically deduce the relation of natural order on R from the field structure; in this way, the order relation has a derived status in the axiomatic system. In fact, it is definable; and, since the axiomatic system is built in an algebraic language, it is algebraically definable. If we take L = {0, 1, +, ., =}, we can write 'x > z' knowing that, by definition, this statement is equivalent to 'x - z > 0' and that, in turn, this is equivalent, by virtue of the axiom (2), to '3y (x - z = y.y = y. Remarkably, in this way we rediscover for R a situation identical to that in the ring Z of integers. Indeed according to Lagrange's theorem of 1770, every positive integer can be written as the sum of the squares of four integers. As a consequence the sentence 'an integer is positive if and only if it may be written as a sum of squares' is true. The logical significance of this sentence is that the order relation on Z is definable on the basis of addition and multiplication.9 The progress achieved since Dedekind is considerable: the latter had reduced continuity to order, Artin and Schreier reduced order to calculation. In other words, they built the algebraic theory of inequalities and at the same time provided one of the first examples of the study of an ordered algebraic structure.'l This result initially made a deep impression because of its epistemological import, as we have seen through the reactions of Helmut Hasse and Andr6 Weil. Somewhat later, the logical theory of definability [6, 25, 26] made it possible to give formal expression to this result, and led to a more precise understanding of it. Still later, at the end of the 1960s, certain studies incorporating results from model theory clarified the relationship between order and metric properties (between the properties of being real closed and metrically complete). For example, Kurt Hauschild [10] has shown that the Cauchy-completion of a field K is real closed if and only if it satisfies a statement substantially similar to I.V.T. For his part, Dana Scott [221 has shown that the Dedekind-completion of a field K is real closed if and only if K is dense in its real closure, i.e. in the smallest real closed algebraic extension of K. However, completion alone, whether it be in the Cauchy sense or in the Dedekind sense, does not imply that K is real closed, and the converse is not true either. Obviously there are real closed fields, RA for example, which are neither Cauchy-complete nor Dedekind-complete. Conversely, there are complete fields which are not real closed; the classical example is provided by the completion of the field Q(t) of rational functions with rational coefficients

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ordered by the non-Archimedean order obtained by taking t > q for every q E Q (the proof applies Dana Scott's abovementioned result by showing that Q(t) is not dense in its real closure). It follows that the algebraic property of maximality (real closed field) does not coincide at all with the metrical property of maximality (complete field). However, (1) one can content oneself with the algebraic property, even for the treatment of questions which involve a metric, if one restricts oneself to objects which are elementarily definable, for example the subsets given by a system of sign conditions on a finite number of polynomials (such sets are termed 'semi-algebraic'). In general, in a maximal ordered field one can pose and solve the question of determining which geometrical constructions can be realized by finite algebraic constructions. And (2) it is possible to pass from one property to the other under certain conditions, which can themselves be varied. For example, by using the work of Dana Scott, McKenna [18] has obtained the following result: a field K, ordered and metrically complete, is real closed if and only if it admits an affirmative solution to the seventeenth problem of Hilbert. Thus, several properties can serve to bridge the gap between metric theory and algebra or arithmetic: the I.V.T. whose importance was recognized ever since the start of the theory of equations; the density of a field in one of its extensions and the decomposition of a non-negative element into a sum of squares, whose recent highlighting goes back respectively to Dedekind and Hilbert. As the order relation(s) achieved greater autonomy with respect to metric theory, the more their fate became linked to that of arithmetic. To tell the truth, this connexion is already present in axiom (1) which defines the 'reality' of a field, i.e. the possibility of ordering it, by a condition concerning the sums of squares of elements of K equivalent to this: n

IX, = 0 <- Xi = 0 i

for each i and each n-tuple

(XI,

x 2 , ....

xn) of elements of K.

Another way of formulating this condition is : no quadratic form of 1) on K has a nontrivial solution in K. However, the the type (1..... operational potential of this connexion between the existence of an order on K and the property, termed 'anisotropy', of certain quadratic forms on K lay fallow until the end of the 1960s. It was at that time that the parallel developments in real algebra and in the algebraic theory

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of quadratic forms brought out the close connexions between ordering and calculation, or, to put it otherwise, between number fields and orderable fields. The notion of signature, initially defined by Sylvester for a quadratic form with coefficients in the field of real numbers [24, article Inertia, p. 583], was generalized to any real or real closed field [5, 17, 19]. Thus, Pfister was able to formulate a local/global principle for a real field K analogous to that of Minkowski-Hasse" for algebraic number fields finitely generated over Q. This principle gives an algebraic charaterization of those forms whose signature is zero for any compatible order P of K; the real closures of K, relative to each order P, play the role held by the 'local fields' (Qp or KO)"2 in number theory ([19], Satz 22). Subsequently, the notion of signature was applied not only to quadratic forms but to the fields themselves: a signature of a field K is defined as an epimorphism from the Witt ring W(K) onto the ring Z." The existence of an obvious bijective mapping from the set of all compatible orderings of K onto the set of all signatures of K (or set of all epimorphisms W(K) --* Z) allows, in the case of real fields which are not necessarily real closed, an abstract algebraic interpretation of the notion of an order. More precisely, it has been shown [17] that there exists a bijection from the set of all orders of K onto the set, called 'spectrum', of all minimal prime ideals of the ring W(K). This bijection makes possible to translate any consideration concerning orderings of K into a consideration concerning the Witt ring of K, and hence to treat real fields in the language of the algebraic theory of quadratic forms. For example, if K is a field ordered by an order P and L = K('-a) for a belonging to the positive cone P, then it may be seen that P can be extended into an order P' of L by considering the canonical homomorphism 0: W(K) -- W(L). *

Let us conclude with some remarks on the notion of order. Like other fundamental notions, this one is actually situated upstream of the great divide of mathematical work into geometry versus algebra, topology versus arithmetic, the continuous versus the discrete. Thus, some see the study of order as a strictly geometrical exercise. For example, renewing the Leibnizian tradition of Analysis situs at the end of the last century, Camille Jordan described his research in 'pure geometry' as deepening the theory of order. From this point of view, order seems to be indissolubly linked to the geometrical continuum understood as a

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space which is connected and compact (at least locally). Others see the study of order as a specific expression of the analytical spirit. For example, in his Cours of 1968 Jean Dieudonn6 explicitly defines the infinitesimal calculus as "an apprenticeship in handling inequalities", an apprenticeship which can be summed up in three words: "minorizing, majorizing, approximating". Such activity presupposes a metric space endowed with the Dedekind property or with that of Cauchy; and a set of numbers, the rationals, which is dense in that space. Others still relate the manipulation of inequalities to algebra. Claude Chevalley wrote concerning Weierstrass' definition of the notion of limit: "the great advantage of the new definition of limit . . has been to replace the old passage to the limit ... by an algebraic procedure based on transformations of inequalities" [7, p. 378]. This was in 1935, after the theory of Artin and Schreier (which Chevalley knew well, having resided in Hamburg during the 1930s) had made possible the rigorous handling of inequalities on a purely algebraic basis. A priori or by nature, the notion of order in mathematics is neither intrinsically geometrical, even though it is easily represented by the 'betweeness' relation, nor purely algebraic, even though it can be translated into algebraic inequality, nor yet analytical even though it is implied in the notions of limit and convergence. It is a notion which lends itself to various costumes, to a variety of expressions; in fortunate cases, one manages to show that the different expressions correspond to each other or even that they are equivalent. In contemporary mathematics the notion of order thus appears as a transversal notion, present on many of the paths which lead from one discipline to another. ACKNOWLEDGEMENTS

I would like to thank John Stewart (CNRS, Paris) for his translation, Max Dickmann (CNRS-Universit6 Paris VII, Paris) and Paul Pedersen (Courant Institute, New York University) for their many improvements. Directeur de recherche au CNRS, Institut d'Histoire et Philosophie des Sciences et des Techniques, 13, rue du Four, 75006 Paris,France.

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* A French and slightly different version of this paper appeared in J.-M. Salanskis and Hourya Sinaceur (Eds.), Le Labyrinthe du continu, Springer-Verlag, France, 1992. SFor a detailed study, see [23], especially Part 2. 2 Up until now, the texts of Artin and Artin and Schreier concerning real algebra have only been available in the German original. A joint French-English translation will soon be available [21]. 3 The most famous theorem (1888) states that any system of invariants has a finite base. The proof of this existential sentence is general (non-constructive), but its content corresponds in a way to Hilbert's strategy: to manage the infinite on the ground of the finite (see H. Sinaceur, 'Du formalisme I. la constructivitt: le finitisme', Revue Internationale de Philosophie n' 4/1993, pp. 251-283). In the same spirit, Artin proved some properties of 'artinian' rings, i.e. those rings in which any decreasing chain of ideals is finite. 4 That is to say, statements which can be formulated in a first-order logical language. Of course, the concept of 'elementary statement' is absent in the text of Artin and Schreier. SThe expression 'real number' seems to have come into common usage only after Dedekind and Cantor provided a correct definition of the set of real numbers at the end of the 19th century. Dedekind himself, in [8], still used the expression 'irrational number'. 6 Neither the first nor the third of the axioms can be written directly as a single

elementary statement. However, n-tuple (x,, x 2. 0, X2 + X 0

xi = 0 = x, = 0 for each i, 1 < i < n and for each

x.) of elements of K' is equivalent to the sequence: xý == 0, 0 = X = X = 0.

X2

=

0

==x

=

More precisely: a mapping from one topological space into another is continuous if and only if any neighbourhood of the image of a point contains the image of a neighbourhood of that point. ' In particular, geometrical images or figures. 9 It is not possible to restrict oneself to the operation of addition alone, since it is possible to build a structure ( 7/ Z, +, -) such that the additive group of 7/Zis isomorphic to the additive group of 7/ without 7/Zbeing isomorphic to 7/. "0 Today, the study of ordered algebraic structures (fields, but also rings, groups, etc) constitutes a branch of mathematics in full development. See, for example, [20]. 1 The term 'Minkowski-Hasse principle' refers to the following two statements: (1) two quadratic forms are equivalent over a field K of algebraic numbers, finitely generated over Q, if and only if they are equivalent over the P-adic extensions K, (where 0 is a prime ideal of the ring of integers of K); (2) a form represents zero over K if and only if it represents zero over K, for each

V. 12

up =

{xx=

,•anp Iar #

0, r e 7/, a, e 7/, 0 <_a, < p, p prime numberI or field of

the p-adic numbers K0 = the completion of K (for a certain valuation), where K is an algebraic extension finitely generated over Q. K, is a 'local field', whereas K is a 'global field'. 13 The Witt ring W(K) is defined in [28] as the quotient of the set of the quadratic forms on K by a relation of similarity: (q - q') €* (Ker(q) = Ker(q')).

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205

REFERENCES 1. Artin, E.: 1924, 'Kennzeichnung des Korpers der reellen algebraischen Zahlen', Abhandlungen aud dem mathematischen Seminar der Hamburgischen UniversittIt, 3, 319-323. In [2], pp. 253-257. 2. Artin, E.: 1965, The Collected Papers, edited by S. Lang and J. Tate, Addison, Wesley Pub. Co., 2nd ed., 1982, Springer-Verlag. 3. Artin, E. and Schreier, 0.: 1926, 'Algebraische Konstruktion reeller Korper', Abh. math. Sem. Hamb., 5, 85-99. In [2], pp. 258-272. 4. Artin, E. and Schreier, 0.: 1927, 'Eine Kennzeichnung der reell abgeschlossenen K6rper' Abh. math. Sem. Hamb., 5, 225-231. In [2], pp. 289-295. 5. Becker, E. and Spitzlay, K. J.: 1975, 'Zum Satz von Artin-Schreier Ober die Eindeutigkeit des reellen Abschlusses eines angeordneten Korpers', Commentarii mathematici helvetici, 50, 81-87. 6. Beth, E. W.: 1953, 'On Padoa's method in the theory of definition', Indagationes Mathamaticae, 15, 330-339. 7. Chevalley, Cl.: 1935, 'Variations du style mathdmatique', Revue de Mitaphysique et de Morale, 42, 375-384. 8. Dedekind, R.: 1872, 'Stetigkeit und irrationale Zahlen', Gesammelte mathematische Werke, edited by R. Fricke, 0. Ore and E. Noether, Vieweg, Braunschweig, Vol. III, pp. 315-334. 9. Hasse, H.: 1930, 'Die moderne algebraische Methode', Jahresber.D. M. V., 39, 22-34. 10. Hauschild, K.: 1976, 'Cauchyfolgen hoheren Typus in angeordneten K6rpern', Zeitschrift fuir mathematische Logik und Grundlagen der Mathematik, 13, 5566. 11. Hilbert, D.: 1899, Grundlagen der Geometrie, Teubner; 2nd ed., 1903; 7th ed., 1930; 10th ed. mit Supplementen von P. Bernays, 1968. 12. Hilbert, D.: 1900, 'Ober den Zahlbegriff', Jahresber.D. M. V., 8, 180-184. 13. Hilbert, D.: 1900, 'Mathematische Probleme', in Gesammelte Abhandlungen III, Berlin, 1935. Reprint, Chelsea, New York, 1965, pp. 290-329. 14. Knebusch, M.: 1972, 'On the uniqueness of real closures and the existence of real places', Commentarii mathematici helvetici, 47, 260-269. 15. Lagrange, J. L.: 1798, Traitd de la resolution des equations numiriques de tous les degris. Paris, Duprat, an VI, 2e dd., 1808. Reprint of the 2 ed. in (Euvres, VIII, edited by J. A. Serret and G. Darboux, Gauthier-Villars, Paris, 1867-1892. 16. Laugwitz, K.: 1973, 'Bemerkungen uber Stetigkeit und angeordnete Koper', Demonstratio mathematica, VI, Part 1, 191-209. 17. Leicht, J. and Lorenz, F.: 1970, 'Die primideale des Wittschen Ringes', Inventiones mathematicae, 10, 82-88. 18. McKenna, K.: 1975, 'New facts about Hilbert's seventeenth problem', in A. Dold and B. Eckmann (Eds.), Model Theory and Algebra. A Memorial Tribute to Abraham Robinson, L. N. M. 498, Springer-Verlag, pp. 220-231. 19. Pfister, A.: 1966, 'Quadratische Formen in beliebigen Korpern', Inventiones mathematicae, 1, 116-132. 20. Priess-Crampe, S.: 1983, Angeordnete Strukturen: Gruppen, Korper, projektive Ebenen. Ergebnisse der Math. u. ihrer Grenzgebiete, 98, Springer-Verlag

206 21. 22.

23. 24.

25.

26. 27.

28.

HOURYA SINACEUR Roy, M.-F. and Sinaceur, H.: (to appear), A Source Book of Real Algebraic Geometry, Springer-Verlag, France. Scott, D.: 1969, 'On completing ordered fields', in W. A. J. Luxemburg (Ed.), Applications of Model Theory to Algebra, Analysis and Probability (Proc. int. symp. Institute of Technology, California), Holt-Rinehart & Winston, New York, pp. 274-278. Sinaceur, H.: 1991, Corps et modkles, Essai sur I'histoirede l'alg~brerjelle, Librairie philosophique J. Vrin, Paris. Sylvester, J. J.: 1953, 'On a theory of the syzygetic relations of two rational integral functions, comprising an application to the theory of Sturm's functions, and that of the greatest algebraical common measure', Phil. Trans. Roy. Soc. London, 143, 407-548. Reprint in The Collected Mathematical Papers,Cambridge University Press (1904), Vol I, pp. 429-586. Tarski, A.: 1931, 'Sur les ensembles d6finissables de nombres r6els F', Fundamenta mathematicae, 17, 210-239. Reprint in Collected Papers, edited by S. R. Givant and R. N. McKenzie, Birkhauser (1986), Vol. I, pp. 517-548. Tarski, A.: 1935, 'Einige methodologische Untersuchungen uiber die Definierbarkeit der Begriffe', Erkenntnis, 5, 80-100. Collec. Papers, Vol. II, pp. 637-659. Weil, A.: 1967, 'Review of the Collected papers of Emil Artin', Scripta Mathematica, 28, 237-238. Reprint in (Euvres scientifiques, 2nd ed., Springer-Verlag (1980), Vol. III, pp. 173-174. Witt, E.: 1937, 'Theorie der quadratischen Formen in beliebigen Korpern', J. r. angew. Math., 176, 31-44.

H. JEROME KEISLER

THE HYPERREAL LINE

1.

INTRODUCTION

The aim of this article is to explain that the hyperreal line is, what it looks like, and what it is good for. Near the beginning of the article we shall draw pictures of the hyperreal line and sketch its construction as an ultrapower of the real line. In the middle part of the article, we shall survey mathematical results about the structure of the hyperreal line. Near the end, we shall discuss philosophical issues concerning the nature and significance of the hyperreal line. The hyperreal number system is a very rich extension of the real number system which preserves all first-order properties. It contains infinite and infinitesimal numbers. Another important feature is the 'hyperfinite grid', which is an infinite set of equally spaced points on the unit interval which has the first-order properties possessed by all sufficiently large finite grids. We begin with a brief history of the hyperreal line. Archimedes (287-212 B.C.) in his manuscript 'The Method' introduced infinitesimals as a method for the discovery of mathematical results. The basic rules of the calculus were discovered in the 1500s and 1600s by reasoning informally with infinitesimals. The method remained dominant until the middle of the seventeenth century, and was the motivation for Leibniz' differential notation dx. Leibniz (1646-1716) correctly anticipated the modern viewpoint; he regarded the infinitesimals as ideal numbers like the imaginary numbers, and proposed his law of continuity: "In any supposed transition, ending in any terminus, it is permissible to institute a general reasoning, in which the terminus may also be included." This 'law' is far too imprecise by present standards, but was a forerunner of the modern Transfer Principle that the hyperreal number system has the same first-order properties as the real number system. Bishop Berkeley effectively criticized the logical inconsistencies in the intuitive use of infinitesimals in 1734. A precise treatment of either the real or the hyperreal number system was beyond the state of the art at the time. In the 1870s set theory was developed by Cantor, Bolzano, 207 P. Ehrlich(ed.), Real Numbers, Generalizationsof the Reals, and Theories of Continua, 207-237. © 1994 Kluwer Academic Publishers. Printedin the Netherlands.

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Dedekind, and Weierstrass to the point where the real line could be precisely defined, and Weierstrass based the classical rigorous treatment of the calculus on the real line. A mathematically precise construction of a hyperreal line had to wait for developments in the area of mathematical logic, in particular model theory. To construct the hyperreal number system, one needs either the compactness theorem or some version of the ultrapower construction. In 1927, Thoralf Skolem introduced the ultrapower construction to obtain a nonstandard model of the first-order theory of arithmetic. The compactness theorem for countable languages was proved by Gbdel in 1930, and the general case of the compactness theorem was proved by Malcev in 1936. In 1948, Hewitt used the ultrapower construction, from the point of view of algebra rather than logic, to obtain a real closed extension of the ordered field of real numbers, and introduced the name hyperreal numbers. In 1955, -Log introduced the ultrapower construction in general and proved the Transfer Principle that the ultrapower preserves all first-order properties. The beginnings of an infinitesimal approach to analysis was developed in 1958 by Laugwitz and Schmieden (Laugwitz and Schmieden, 1958). The big step came in 1960, when Abraham Robinson realized that the Transfer Principle could be used to give a mathematically correct development of all of analysis based on the hyperreal number system, and gave the method the name nonstandard analysis. Robinson's early papers and his book (Robinson, 1966) developed the hyperreal number system within the theory of types. The ultrapower treatment was popularized in the monograph by Luxemburg (1962). Since then, it has become evident that the hyperreal number system is a tool which is broadly applicable in many areas of pure and applied mathematics. For the last century, the mathematical community has had a strong aversion to the notion of an infinitesimal. The original reason for this aversion is the historical fact that infinitesimals were used incorrectly in the development of the calculus, and the errors were corrected by the treatment of Weierstrass based on the real line (banishing infinitesimals). More recently, the influence of the hyperreal line on mathematics has been held back by the fact that mathematical logic has seldom been regarded as an essential part of the graduate mathematics curriculum. Although it is formally possible to develop the hyperreal number system without mathematical logic, one must be comfortable with mathemat-

THE HYPERREAL LINE

209

ical logic in order to make effective use of the hyperreal numbers as a research tool. Some recent books on the hyperreal line and its role in mathematics are (Albeverio et al., 1986; Cutland, 1988; Hurd and Loeb, 1985; Robert, 1988; Strogan and Bayod, 1986). For an elementary treatment at the freshman calculus level, see (Keisler, 1986). For background material and references in the theory of models, see (Chang and Keisler, 1990). The work was supported in part by the National Science Foundation and the Vilas Trust Fund.

2.

INFINITESIMALS

One of the most striking and best known features of the hyperreal line is the presence of infinitesimals and infinite numbers. However, infinitesimals are only part of the story. They arise not only in the hyperreal line but in much simpler structures, namely, in any ordered field which does not have the Archimedean property. In the beginning stages of Robinson's analysis, only a few algebraic rules are needed, and one can go surprisingly far into the infinitesimal calculus of algebraic functions with such rules. More power is not required until one gets to the transcendental functions. These rules are contained in the notion of a non-Archimedean real closed ordered field. In this section and the next shall discuss the picture of the line from a non-Archimedean viewpoint. We begin with the basic definitions. DEFINITION. Given an ordered field F = (F, +, -, 0, 1, <), we shall identify each natural number n c Ni with the field element 1 + . . . + 1 (n times). An element x of F is said to be infinitesimal iff lxi -< 1/n for all positive integers n, finite iff lxi < n for some integer n, and infinite iff lxi ->n for all integers n. Two elements x, y are said to be infinitesimally close, in symbols x = y, iff y = x + d for some infinitesimal d. DEFINITION. An ordered field is said to be non-Archimedean iff it has at least one positive infinite element, and Archimedean otherwise. A fundamental classical result is that an ordered field is Archimedean if and only if it is isomorphic to an ordered subfield of the field R of

210

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real numbers. For example, the ordered fields of real numbers, rational numbers, and real algebraic numbers are Archimedean. Examples. Here are some examples of non-Archimedean ordered fields which are defined without recourse to methods from mathematical logic. Each example except (c) is an extension of the field of real numbers. (a) The field R(x) of all rational functions over R of one variable x (i.e. quotients of polynomials in x with real coefficients), with the ordering determined by stipulating that x is positive infinite. (b) The real closure of R(x), that is, the smallest ordered field containing R(x) which is closed under taking roots of polynomials of odd degree and square roots of positive elements. (c) The field Q(xl ..... xj)of rational functions over the rationals 0 in n variables, with the ordering determined by making x, positive infinite and xi,, greater than any power of xi for i = 1.... n - 1. (d) The field of Laurent series, i.e. formal power series of the form I-k anx with real coefficients and k r= N, making x positive infinite. (e)

The ordered field of surreal numbers of Conway [Co].

Another example of a non-Archimedean ordered field is the ordered field of hyperreal numbers, which will be introduced in due course. BLANKET ASSUMPTION. Throughout this and the next section, IFwill denote a non-Archimedean ordered field which contains the real number field R. PROPOSITION 1. (Algebra of infinitesimals). (i) The finite elements of Fform a convex subring. That is, sums, differences, and products of finite elements are finite, and any element between two finite elements is finite. (ii) The infinitesimal elements of IFform a convex ideal in the ring of finite elements. That is, sums and differences of infinitesimals are infinitesimal, the product of an infinitesimal and a finite element is infinitesimal,and any element between two infinitesimals is infinitesimal. (iii) If H is positive infinite, then each K > H is positive infinite, H + x is positive infinite for all finite x, and H . y is positive infinite for all positive noninfinitesimal y.

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(iv) If x is nonzero infinitesimalthen 1/x is infinite, and if y is infinite then 1ly is infinitesimal. We can see now that a non-Archimedean ordered field has infinitely many different infinitesimals and infinite elements. By definition, there is at least one positive infinite element H. By (iii) there are infinitely many different positive infinite elements. By (iv), d = 1/H is positive infinitesimal. By (i), all finite multiple of d, all finite powers d" of d, etc. are infinitesimal. Given any positive infinitesimal c, -c is a negative infinitesimal. The set of infinitesimals may be thought of as a small cloud centered at zero, and the set of finite elements a large cloud centered at zero. About each x E F, there is another small cloud, called the monad of x, consisting of the set of all elements which are infinitesimally close to x, and a large cloud, the galaxy of x, consisting of the set of all elements which are at a finite distance from x. The monad of zero is the set of infinitesimals, and the galaxy of zero is the set of finite elements. Note that each galaxy is a union of monads. 3.

STANDARD

PARTS AND ELEMENTARY EXTENSIONS

We shall now look at the real number field R and a non-Archimedean extension F as a pair of structures. We introduce the standard part function, which associates a real number with each finite element of F. DEFINITION. Let x be a finite element of F. A real number r is said to be the standard part of x, in symbols r = st(x), if r is infinitesimally close to x. Infinite elements do not have standard parts. PROPOSITIONS 2. Each finite element of F has a unique standardpart. Hint. Show that if x e F is finite, then the least upper bound of the set of all real r < x is the standard part of x. Propositions 1 and 2 provide us with the picture, shown in Figure 1, of a non-Archimedean line floating above the real line. In the normal real scale without magnification, the non-Archimedean line looks like the real line. But if we train an 'infinitesimal microscope' on a finite element x, and set the magnification to be an infinite factor y, the monad of x will be infinitely magnified and points which

212

H. JEROME KEISLER

ir;11

neg. infinite

mirrnPnnp

finite

inf'l microscope

J•

•J

0

1

infinite telescope

7

pos.

infinite

REAL LIHE

Fig. 1. The real and non-Archimedean Lines.

are at the infinitesimal distance 1/y apart will be distinguishable. If we aim an 'infinite telescope' at a positive infinite element, we will see positive infinite elements in the normal real scale. An ordered field is said to be real closed iff every positive element has a square root and every polynomial of odd degree has a root. Among the examples in the preceding section, (b) and (e) are real closed ordered fields. The ordered fields of real numbers, real algebraic numbers, and hyperreal numbers are also real closed. In the classical paper (Tarski and McKinsey, 1948), Tarski proved that all real closed ordered fields have the same properties which are expressible in first-order logic. More precisely: PROPOSITION 3. (i) The first-ordertheory of real closed orderedfields is complete, that is, all real closed ordered fields satisfy the same sentences of first-order logic. (ii) The first-order theory of real closed orderedfields is model complete, that is if G C F are real closed orderedfields, then any formula of first-order logic with names for elements of G is true in F if and only if it is true in G. In particular, if F is a real closed ordered field which contains R, then any first-order formula with names for real numbers is true in F if and

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only if it is true in the real numbers. In model theory, this is expressed briefly by saying that IFis an elementary extension of R. There is an important loophole in the result of Tarski. It applies only to formulas in the language of ordered fields, that is, formulas built up from the predicates =, <, the function symbols +, ., and the constants 0 and 1. Nothing has been said about how one might extend other real relations or functions to the larger set F. For example, the exponential functions, the trigonometric functions, and the set of natural numbers cannot be defined in the language of ordered fields. One of the main properties of the hyperreal line will be the Transfer Principle, which says that the hyperreal number system is an elementary extension of the real number system even in the full language which has a symbol for every relation and function over R. 4.

THE HYPERREAL NUMBERS

The hyperreal number system can be introduced either by an axiomatic approach which lists its properties and proves that a structure with those properties exists, or by an explicit construction. In this article we shall construct the hyperreal number system explicitly as an ultrapower of the real number system. This has the advantage of giving the reader a concrete structure, but the disadvantage of being less natural than the axiomatic approach. Moreover, the ultrapower construction is special, in the sense that not all hyperreal number systems allowed by the axiomatic approaches are ultrapowers of the real number system. In Section 12 we shall touch on two axiomatic approaches which extend not only the real line but the whole set theoretical universe in which the real line lives. We now define the notion of an ultrapower of the real number system R. Let I be an infinite set, the index set. Let us consider a family U of subsets of I. An ultrafilter over I is a set U of subsets of I such that (1) IfXE UandXC YCI, then YE U; (2) If X, YE UthenXfl YE U; (3) For all X C I, exactly one of the sets X, I\X belongs to U. Given an ultrafilter U, we shall form a new structure *R, called the ultrapower of R modulo U. In order to see how the conditions (1)-(3) arise naturally, let us temporarily consider an arbitrary set U of subsets of I. We begin with the set R' of all functions from I into R. Our plan is to identify certain elements of R' .with each other, so the elements of

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H. JEROME KEISLER

the ultrapower will be equivalence classes of elements of R'. We first deal with the properties of equality and order, and introduce two relations =u and
*R = {fu : f E R'}. We identify each real number r E R by the equivalence of its constant function, r = (I x {rl)u, so that R C_ *R. If R is a proper subset of *R, we call the ultrapower nontrivial. The following proposition, which depends on the axiom of choice, insures the existence of nontrivial ultrapowers. PROPOSITION 5. (i) (Tarski, 1930). Every infinite set has an ultrafilter which is not closed under countable intersections. (ii) The ultrapower of R modulo U is trivial if and only if U is closed under countable intersections. We now define relations and operations on *R, by generalizing the definition of =u and <5u. Let Y, be the set of all relations and functions on R. By the full structure over R we mean the model (R, S : S E fl. We shall expand *R to a model for the vocabulary 9;. Let S be an n-ary relation and G an n-ary function on R. For all f, . . . fn, E R', we define

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215

*S(fil ... fIL) iff {i e I: S(fl(i) ..... f"(i)) E U), *G(flu ..... f~u) = (G(fU(i)... f-(i)) : i C I)U. That is, a relation holds in the ultrapower iff it holds for U- almost all i, and a function in the ultrapower is defined component-wise. It can be checked that these are proper definitions, that is, they depend only on the equivalence classes of fl, . . . , f, and not on the functions themselves. We are now ready to state our definition of a hyperreal number system. DEFINITION. By a hyperreal number system we shall mean a structure of the form (*R, *S : S E 9,) where *R is a nontrivial ultrapower of the real line R. By a hyperreal line we mean the ordered set (*R, *<) in a hyperreal number system. We shall assume hereafter that *R is a nontrivial ultrapower of R. Relations and functions over R will be called real, and stars of real relations and functions will be called standard. Thus a relation T over *R is standard iff T = *S for some real relation S. For example, the set N of natural numbers is a real set, and the set *N of hypernatural numbers is a standard set. The usefulness of the hyperreal number system is based on two fundamental properties, the Transfer and Saturation Principles. The Transfer Principle formalizes the intuitive idea that the hyperreal number system should be as much like the real number system as possible. The Saturation Principle, stated in the next section, captures the fact that the hyperreal number system is very rich. TRANSFER PRINCIPLE (Log 1955). Any hyperreal number system (*R, *S : S E f) is an elementary extension of the full structure (R, S : S E f). That is, every n-tuple of real numbers satisfies the same first-order formulas in (R, S : S E 9) as it satisfies in (*R, *S: S E The proof is by induction on the complexity of formulas. It follows from the Transfer Principle and the fact that *R is a proper extension of R that *R is a non-Archimedean real closed ordered field. Moreover, all other real functions, such as the exponential and trigonometric functions, satisfy the same inequalities in the hyperreal number system that they satisfy in the real number system.

216

H. JEROME KEISLER 5.

INTERNAL SETS AND SATURATION

The concept of an internal set is of overriding importance in Robinson's analysis. Intuitively, the internal sets are those sets of hyperreal numbers which inherit the first-order properties of sets of the real numbers. Actually, we work with n-ary relations rather than only sets. Ordinarily, the internal relations are defined as those relations which are ultraproducts of real relations. We shall give a simpler but slightly stronger definition here which will be sufficient for our purposes; our internal relations will just be the sections of standard relations. We use the vector notation a for an n-tuple (a ..... a,.). DEFINITION. An n-ary relation Ton *R is said to be internaliff there is a real (n + 1)-ary relation S and a hyperreal number b such that T = {a e *R' : *S(a, b)}. Remark. It can be shown that any relation which is internal in the above sense is an ultraproduct modulo U of a family of standard relations. The converse also holds if the index set I has cardinality at most 2. The next two results are useful consequences of the Transfer Principle. THE OVERSPILL PRINCIPLE. Every nonempty bounded internal subset of *R has a supremum in *R. Hint. Apply Transfer to the sentence which states that every nonempty bounded section of a relation has a supremum. Examples. By the Overspill Principle, The following subsets of *R are not internal, since they are nonempty and bounded but have no supremum: N, R, the monad of 0, the galaxy of 0. As these examples show, the hyperreal line is not completely ordered. This example contains a warning: the Transfer Principle does not hold for second-order formulas. The Overspill Principle is used in almost every application of the hyperreal number system. A typical exercise: Use the Overspill Principle to show that for each real function f, lim f(x) = 0 if and only if *f(x) - 0 for all positive infinite x e *R. In order to make effective use of the hyperreal number system, it is

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vitally important to be able to recognize an internal relation. In practice, we usually use the following principle to show that a relation is internal. It is a corollary of the Transfer Principle. INTERNAL DEFINITION PRINCIPLE. Let 4(x, y) be a formula in the vocabulary 9;, and let b be a tuple of hyperreal numbers. Then the relation {a

E

*R' : 0(a, b) holds in (*R, *S : S e 3)l

is internal. Example. (a) Any standard relation *S over *R is internal. (b) The collection of internal sets is closed under finite unions, finite intersections, and complements. (c) If F is an internal function, then the domain and range of F are internal. (d) For each pair of hyperreal numbers c, d, the *closed interval [c, d] = {aE *R : c * a and a *< d} is internal. (e) For each hypernatural number H E *N, the set {10...,H-1)={KE *N :K*
218

H. JEROME KEISLER

(b) No countable strictly increasing sequence of hyperreal numbers has a least upper bound. (c) Any countable set of infinitesimals has an infinitesimal upper bound. Any countable set of hyperreal numbers has a hyperreal upper bound. 6.

THE HYPERFINITE GRID

We now introduce the hyperfinite sets. Intuitively, a hyperfinite set is a subset of *R which inherits the first order properties of finite subsets of R. By analogy with the internal sets, a hyperfinite set is ordinarily defined as an ultraproduct of a family of finite sets. As in the case of internal sets, we shall use a simpler but slightly stronger definition; our hyperfinite sets will be the sections of standard relations *S such that every section of the original real relation S is finite. DEFINITION. A subset T of *R is said to be hyperfinite iff there is a real relation S such that {a E R : S(a, b)} is finite for all b E R and T

=

{a E *

*S(a, b)}) for some b E * R.

Clearly, each hyperfinite set is internal, and each finite subset of *R is hyperfinite. An example of a hyperfinite set which is infinite is the set {0 .... H - 1 I where H is a positive infinite hypernatural number. By the Transfer Principle, a set T is hyperfinite if and only if there exists a hypernatural number H and an internal bijection F from {0 ..... H - 1 } onto T; we call H the internal size of T. Moreover, any internal subset of a hyperfinite set is hyperfinite. By writing integers in base 2 and using Transfer, a hyperfinite set X C_ *N may be coded by the hypernatural number c(X) = Y-{2J : J E X}. This coding reduces properties of hyperfinite subsets of *N to properties of elements of *N, and is useful in combination with the Internal Definition Principle. PROPOSITION 6. Here are some consequences of the Transfer and Saturation Principles which help to give us a picture of the hyperfinite sets.

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(i) (ii)

Any nonempty hyperfinite subset of *N has a greatest element. Given an increasing chain S, and a decreasing chain T" of hyperfinite sets such that S_ C T. for all n E N, there exists a hyperfinite set X such that S, C X C T, for all n ( N. Hint. Use part (ii) and Saturation. (iii) The union of a countable strictly increasing chain of hyperfinite sets is external. Hint. Similar to (ii). (iv) Any function f: N -- *N can be extended to an internalfunction F : *N -- *N. Hint. Use part (ii), Saturation, and the coding c(X). One of the most important features of the hyperreal line is that each interval can be partitioned into hyperfinitely many subintervals of the same length. This feature is captured by the notion of a hyperfinite grid. DEFINITION. Choose a positive infinite hypernatural number H. By the hyperfinite grid with mesh 1/H we mean the hyperfinite set of all multiples of 1/H between -H and H, H = {K/H : K

E *"

and IKI < H2}.

We think of H as the set H

1-HH+-H+ 2 ............................

S1 2 01 H'I'HHH...

2 H'

2 1 H , - H' ,H --- H'H

1 H' .

We usually take H so that every standard natural number divides H. It then follows that each standard rational number belongs to H, that is, QCH. The notions of an interval, monad, and galaxy in H are defined by restricting the original notions to H in the obvious way. The hyperfinite grid is not a field, and in fact is not even closed under addition, subtraction, multiplication, or division. The coarse picture of the hyperfinite grid as an ordered set is much like the hyperreal line or non-Archimedean line, with a monad surrounding each real number, a finite part, and a negative and positive infinite part. However, the finer details, shown in Figure 2, are markedly different. A hyperfinite grid H has a least and a greatest element and is discretely ordered,that is, each element except the least has an immediate

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inf'l microscope

nee,.

infinite

finite

inf'l microscope

infinite telescope

•st

,

0

pos. infinite

1

PEAL LINE Fig. 2. The real line and the hyperfinite grid.

predecessor and each element except the greatest has an immediate successor. If we train an infinitesimal microscope on an element of the hyperfinite grid and set the magnification to H, the portion of the grid inside the microscope will look like a portion of the set Z of integers. If we train a microscope on the greatest element H and aim the telescope at this microscope, we will see the upper end of the hyperfinite grid, which will look like a portion of the set of negative integers with 0 at H. The hyperfinite grid, like any discretely ordered set with a first and last element, many be divided into a first block which is a copy of the positive integers, a linearly ordered set of blocks which are copies of the set Z of integers, and a last block which is a copy of the negative integers. Moreover, the ordering of the blocks is dense, because if x and y are in different blocks, then 1/2(x + y) is in a block strictly between the two. The set of real numbers R is not contained in H. In fact, H1 is a subset of the set *Q of hyperrational numbers, and no irrational number belongs to *Q. On the other hand, every real number is infinitely close to some element of H, so the standard part function maps the finite part of H onto R. Hint. Let r be real. By Transfer, there is a greatest element [r] of H such that [r] < r, and [r] is infinitely close to r.

THE HYPERREAL LINE 7.

221

THE LOEB MEASURE

One of the benefits of the hyperfinite grid is that one can calculate with it as if it were finite. Such calculations are legal only for internal objects. The internal subsets of the hyperfinite grid are the 'simplest' subsets; they are themselves hyperfinite, and play a role somewhat like the intervals, or finite unions of intervals, on the real line. A common theme is to approximate a set in which one is interested by an internal set. In this section we illustrate this theme with the Loeb measure construction, which has proved to be an extremely powerful tool. The idea is to assign an equal infinitesimal weight 1/H to each element of the hyperfinite grid, compute an internal measure which assigns a hyperreal value to each internal subset by adding up weights, take standard parts to get a real valued measure, and then use Saturation to extend to a countably additive measure in the classical sense. We choose the weight 1/H because then for any a, b r- H, the internal measure of an interval H f [a, b) will be equal to the length b - a. Thus the internal measure of an interval of infinite length will be infinite. To avoid side issues involving sets of infinite measure, from now on we shall concentrate on the hyperfinite unit interval H, consisting of all x E H such that 0 < x < 1. DEFINITION. Let A be an internal subset of H,. By the internal counting measure of A we shall mean the hyperreal number Rt(A) = #(A)/H, where #(A) is the internal size of A. By the Loeb measure L(g)(A) of A we mean the standard part st(g(A)). The set of Hl, itself has internal size H and thus has internal counting measure 1. Recall that the set of internal subsets of H, is a field of sets but is not countably additive. At this point we have only defined the Loeb measure for internal sets. This restriction of the Loeb measure is finitely additive. Our next step is to assign Loeb inner and outer measures to arbitrary subsets of H,. DEFINITION. Let B be a subset of H,. The inner Loeb measure of B, Ptinne.(B), is defined as the supremum (in the classical real sense) of the Loeb measures of all internal subsets of B. The outer Loeb measure of B, .outer(B), is the infimum of the Loeb measures of all internal supersets of B.

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Finally, we define the Loeb measure and show that it is countably additive. DEFINITION. A set B C H-,l1is said to be Loeb measurable iff li-tnn.r(B) = [tour(B). If B is Loeb measurable, its Loeb measure is the common value of its inner and outer measure. PROPOSITION 7. (Loeb, 1975). The Loeb measure is a complete countably additive measure. Hint. The key step is to show that a countable union of sets B. of outer Loeb measure zero has outer Loeb measure zero. Given a real S> 0, for each n we may choose an internal superset A. D B_ of counting measure < r2'. Now use Saturation to get a single internal set of counting measure < 2F which contains all of the sets An. Another nice consequence of Saturation is the 'internal approximation lemma', that for every Loeb measurable set B there is an internal set A such that the symmetric difference between A and B has Loeb measure zero. There is an elegant relationship between the Loeb measure on the hyperfinite grid and Lebesgue measure on the real line, due to Anderson (1976) and Henson (1979). For any subset C of the real interval [0, 1], C is Lebesgue measurable and only if the set st-'(C) = {x E H, : st(x) e C} is Loeb measurable, and the Loeb measure of st-'(C) equals the Lebesgue measure of C. Here are some examples of sets which are not Loeb measurable (as usual, we omit the proofs). Examples. (a) (Luxemburg) Let A be the internal set of all x e *[0, 1] such that the Hth binary digit of x is 1. Then A n [0, 1] is not Lebesgue measurable so st-'(A n [0, 1]) is not Loeb measurable. (b) The set B = {y e H, : y > st(y)} is not Loeb measurable, but has the measurable standard part st(B) = [0, 1]. Although the Loeb measure lives on a hyperfinite set and is amenable to finite-like computations, it is an extremely rich measure space. This richness has been exploited to prove numerous existence theorems in probability theory. For more about the subject, see the book by Albeverio et al. (1986). The Loeb measurable construction can also be carried out on the whole hyperreal line instead of just the hyperfinite grid. However, the theory

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is less satisfying. On the hyperfinite grid, every point has the same infinitesimal weight and every internal set is Loeb measurable. But on the hyperreal line, each point has weight zero and only some of the internal sets are Loeb measurable. 8.

HYPERFINITE DESCRIPTIVE SET THEORY

Descriptive set theory on the real line deals with the Borel and projective hierarchies of subsets of R•. The open and closed sets are at the first level of the Borel hierarchy, and are called the Y0 and I-I sets, respectively. Given a countable ordinal (x > 0, the Y,' sets are the countable unions of sets in UpJrI, and the f1° sets are the countable intersections of set in U,_J• p. The collection of Borel sets is 0 = sets are obtained from the -l and '°,and the + and fI-I' = U(X1 Y_ relations by existential and universal quantification. There is a parallel descriptive set theory on a hyperfinite grid which begins with the internal sets at the initial level; see Keisler et al. (1989). The Saturation Principle plays a crucial role in the theory and acts as a substitute for the existence of a countable dense set on the real line. This theory has exposed both similarities and differences between the real line and the hyperfinite grid. We give a sampling of results here to build on our picture of the hyperfinite grid. Again, we simplify the discussion by restricting ourselves to the unit intervals H, and [0, 1]. We take the internal subsets of the unit interval lH, in the hyperfinite grid to be both Yo and rl° sets, and form the Borel hierarchy. Each monad is a H1I° set, that is, a countable intersection of internal sets. The sets in the Borel hierarchy are called Loeb sets, and the Loeb sets form the a-algebra generated by the internal sets. Since the Loeb measure is countably additive, every Loeb subset of H, is Loeb measurable. The projective hierarchy is defined in the natural way. A starting point of the theory is the result of Robinson that a set B C [0, 1] is closed if and only if A = st(B) for some internal set B C H1. Kunen and Miller (1983), improving an earlier result of Henson (1979), showed that the standard part inverse preserves the exact location in the hierarchies. That is, a set B C [0, 1] belongs to Yo, or Y,, if and only if st-r(B) belongs to Y_', or Y_, respectively. Given any partial function on the hyperfinite grid whose graph is

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a , or H-', the domain of the function is of the same class. In each case, the analogous statement fails for the real line. The uniformization, or selection, properties for the hyperfinite grid are also different from the corresponding properties for the real line. It follows from Saturation that every X•, or Y0, relation over the hyperfinite grid has a choice function whose graph is 10, or Y-0, respectively. The analogous statements are false over the real line. On the other hand, there is a rlo relation over the hyperfinite grid which has no 1-' choice function for any n. By contrast, every r-[W relation over the real line has a fl'I choice function. A natural and useful notion which has no counterpart in the real line is the notion of a countably determined set (Henson, 1979). A set B C H is said to be countably determined iff B is a finite or infinite Boolean combination of some countable collection of internal sets. All the sets in the projective hierarchy 1-' are countably determined. A convenient way to show that a subset of H, is highly complex is to show that it is not countably determined. For example, no well ordering of H, is countably determined, no function from the unit interval H1 onto the whole grid H is countably determined, and no choice function for the Mz relation in the preceding paragraph is countably determined. 9.

TOPOLOGY AND ORDER ON THE HYPERREAL LINE

The topology on the hyperreal line which has been used most frequently in the literature is the S-topology, where S stands for 'standard'. For simplicity, we shall restrict our attention to the hyperreal unit interval *[0, 1]. A set B C *[0, 1] is open in the S-topology iff for each x E B there is a standard real , > 0 such that each point of *[0, 1] within r of x belongs to B. The S-topology is just the ordinary topology on [0, 1] but replacing each real number by its monad. Robinson's original nonstandard characterizations of limit and continuity are formulated using the S-topology. Thus, a real function f on [0, 1] is continuous if and only if *f is continuous in the S-topology. Robinson introduced another topology on the hyperreal line, called the Q-topology. A set B C *[0, 1] is open in the Q-topology iff for each x E B there is a hyperreal , > 0 such that each point of *[0, 1] within F of x belongs to B. Examples. (a) The internal function *sin(H.x) where H is positive infinite is Q-continuous but not S-continuous.

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(b) The internal 'step' function [x/HI where H is positive infinite is S-continuous but not Q-continuous. We can get more insight into the nature of the hyperfinite line by studying its Dedekind cuts. This leads to a collection of order topologies associated with cuts. The hyperreal line is studied from this viewpoint, for example, in (Zakon, 1969; and Keisler and Leth, 1991); here we shall only mention some simple observations and examples. DEFINITION. By a cut in the hyperreal unit interval *[0, 1] we shall mean a nontrivial initial segment C of *[0, 1] such that C has no greatest element and its complement has no least element. A cut is said to be additive iff it is closed under addition. By the Overspill Principle, there are no internal cuts. A cut C is said to be regular iff for every x > 0 in *R there exists y a C such that x + y 0 C. Zakon (1969) asked whether the hyperreal interval has regular cuts. It was shown by Kamo (1981) that there exist hyperreal lines with regular cuts, assuming the continuum hypothesis. Jin and Keisler (1993) proved this fact in ZFC. Hereafter we shall concentrate on the additive cuts. Clearly, no additive cut is regular. Additive cuts are of special interest because each additive cut induces a topology on the hyperreal line. DEFINITION. Let C be an additive cut in *[0, 1]. By the C-monad of a point x E *[0, 1] we mean the set of all y E *[0, 1] such that Ix - yJ e C. the C-topology on *[0, 1] is defined by calling a set B C *[0, 1] C-open iff for every x E B, there exists E 0 C such that each point of *t0, 1] within e of x belongs to B. The C-topology is not Hausdorff, because two points in the same C-monad belong to the same C-open sets. However, if we identify all the points in the same C-monad, we obtain a Hausdorff topology, which is just the order topology on the C-monads. Examples. (a) The largest additive cut in *[0, 1] is the set of infinitesimals, and the corresponding topology is the S-topology. For any x E *(0, 1], there is a greatest additive cut below x, namely the set C. of all z E *[0, 1] with z/x = 0. The Cx-topology will look like l/x copies of the real interval [0, 1) laid end to end (with real points replaced by Cx-monads).

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(b) If C is an additive cut which is not of the form Cx, the C-topology will be totally disconnected, that is, any two distinct C-monads will be separated by clopen sets. Hint. Given two points x and y in different C-monads, take a b 0 C which is infinitesimal compared to lY - xj and form the clopen set {z : Ix - zj/b is finite. Each of the cuts in examples (c)-(e) below cannot be of the form Cx and therefore induce totally disconnected topologies. (c) For each infinitesimal y r *[0, 1] there is a least additive cut above x, namely the set C of all z ( *[0, 1] such that z/y is finite. (d) Each increasing sequence (xa : (X< %)indexed by a limit ordinal X such that each xa,/xa,, =-0 induces an additive cut C. (e) Lightstone and Robinson (1975) considered the C-topology where C is a cut the form {y : y _<x" for all n E N} for some infinitesimal x. (f) If we relax the requirement that a cut has no greatest element and allow {0} as an additive cut, the corresponding topology on *[0, 13 is the Q-topology. This topology is also totally disconnected. One way to classify cuts is by their cofinality and coinitiality. The cofinality of a cut C is the least cardinality of a subset C which has no upper bound in C. By the coinitiality of C we shall mean the least cardinality of a subset of the complement of C which has no lower bound. The greatest additive cut below x has cofinality 0o, and the least additive cut above x has coinitiality wo.It follows from Saturation that each regular cut has uncountable cofinality and coinitiality. Another consequence of Saturation is that there is no cut which has both cofinality 0o and coinitiality (o. Does there exist a hyperreal line (satisfying the Transfer and Saturation Principles) such that every cut C (or every additive cut C) has either countable cofinality or countable coinitiality? Jin (1992) proved that it is consistent with ZFC that the answer is no. There exist hyperfinite lines with the opposite property, i.e. some additive cut C has uncountable cofinality and uncountable coinitiality. Hint. Take an ultrapower of an ultrapower of R, and consider the cut formed by the second ultrapower applied to the set of finite multiples of 1/H in the first ultrapower. The Baire Category Theorem is an important property of the usual topology on the real interval. The next result shows that on the hyperreal interval, all of the C-topologies satisfy the Baire Category Theorem.

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A set B C *[0, 1] is said to be C-nowhere dense iff each interval of length 0 C contains a subinterval of length o C which is disjoint from B. A countable union of C-nowhere dense sets is said to be C-meager. PROPOSITION 8. (Hyperreal Baire Category Theorem). For each additive cut C in *[0, 1], the set *[0, 1] is not C-meager. The set *[0, 1] is also not meager in the Q-topology. Hint. By Saturation, the intersection of a countable chain of intervals of length 0 C is nonempty. On the real line, there is an interesting interplay between the sets of Lebesgue measure zero and the meager sets. On the hyperreal line there is a similar interplay between sets of Loeb measure zero and C-meager sets. The following examples give some idea of what can happen. Examples. (a) The unit hyperfinite grid H, is C-meager if and only if 1/H 0 C. In the remaining examples, we consider subsets of H, and let C range over the additive cuts such that 1/H E C. (b) The 'Cantor ternary' set, consisting of all x E H I such that x.H has no Is in base 3, is C-meager for all C and has Loeb measure zero. (c) Let P be the set of all x E H, such that x .H is *prime. P has Loeb measure zero. P is C-meager where C is the smallest additive cut greater than I/H, but is not S-meager; in facts, the family of cuts C for which P is C-meager is rather complicated. (d) If C has either confinality o or coinitiality o, then there is a C-meager subset of H, of Loeb measure one. Hint. Find a Cantor set in a hyperfinite base which is C-nowhere dense and of Loeb measure close to one. (The situation for other cuts C is more complex; see Keisler and Leth, 1991.) (e) There is an internal subset A C H, such that A has Loeb measure zero but for all C, A is not C-meager. Hint. Start with a real set B C N such that for each n, B nl [3", 3 "'+)contains one interval of length 2n and all multiples of 2'. 10.

IS THE REAL LINE UNIQUE?

A classical theorem is Zermelo set theory states that there is a unique complete ordered field up to isomorphism. Thus within any model of Zermelo set theory there is a unique real line. To a Platonist who regards

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the axioms of set theory as describing a universe of sets which really exists, the real line is unique. In fact, the unique existence of the real line can be proved in any reasonable set theory which has the power set axiom. We may thus think of the real line as existing uniquely relative to the power set operation. However, there is a case against the view that the real line is unique. Unlike the natural numbers, the real line is not absolute for transitive models (M, E), and different transitive models of set theory have markedly different real lines. By Tarski's theorem, the first-order theory of the real line is the same in all models of Zermelo set theory. However, the second- and higher-order theories of the real line depend on the underlying universe of set theory. The method of forcing has produced innumerable examples of statements in the second-order logic of the real line which are independent of Zermelo set theory and various stronger set theories. Thus the properties of the real line are not uniquely determined by the axioms of set theory. To a mathematician who doubts the existence of the set theoretic universe, or who believes that several competing set theoretic universes exist, the properties of the real line need not be unique. Historically, Zermelo set theory was developed in order to form a rigorous foundation for the real line. A set theory which was not strong enough to prove the unique existence of the real line would not have gained acceptance as a mathematical foundation. The modern set theory KPU, Kripke-Platek set theory with urelements, is an example of a set theory in which the real line cannot be proved to exist. KPU is a natural foundation for recursion theory over sets; a good reference is the book by Barwise (1975). 11.

IS THE HYPERREAL LINE UNIQUE?

The existence and uniqueness of the hyperreal line depends on the choice of the underlying set theory and on how one defines the notion of a hyperreal line. We shall first discuss the situation in ordinary Zermelo or ZermeloFraenkel set theory, with the hyperreal line defined to be an ultrapower of the real line. In ZFC, the existence of the hyperreal number system requires the axiom of choice in two places, first to obtain an ultrafilter which is not closed under countable intersections and then to prove the Transfer Principle (but see Luxemburg, 1962; Pincus, 1974; and Spector,

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1988, for applications and a theory of ultrapowers without the axiom of choice). Given the axiom of choice, there are many different ultrafilters which give rise to difference hyperreal lines. Thus if we take our underlying set theory to be Zermelo or Zermelo-Fraenkel set theory with choice, the hyperreal line can be proved to exist but not to be unique. One can obtain uniqueness up to isomorphism by strengthening ZFC and the definition of a hyperreal number system. By a fully saturated hyperreal number system we shall mean an elementary extension (*R, *S : S E 9) of the full structure over Ii such that any set of fewer than card (*lR) formulas with constants from *R which is finitely satisfiable is satisfiable. It follows from results of the Morley and Vaught (1962) that in ZFC, there exists a unique fully saturated hyperreal number system in any cardinal K > 2o' such that either ic is inaccessible or V = 2• (i.e. the GCH holds at K). In the second case ic = X' = 2X, it follows from (Keisler, 1965) that the fully saturated hyperreal number system of cardinality K is an ultrapower of R. In the case that K is inaccessible, it is open whether a fully saturated hyperreal number system, or even any model of cardinality Kc,can be an ultrapower of R. However, it may not be a good idea to restrict the notion of a hyperreal number system by requiring full saturation. In some applications of the hyperreal number system, full saturation of large cardinality has proved to be a desirable hypothesis. But there are other applications where different hypotheses on the hyperreal number system are needed (for example, it is sometimes useful to take the hyperreal number system to be an ultrapower of R• modulo a selective ultrafilter over N). It is better to leave open the possibility of adding a variety of extra hypotheses on the hyperreal number system. In the case of the real number system, the real line is unique relative to the underlying set theory. The second order theory of the real line is not unique, and this 'absolute' non-uniqueness is sometimes useful. However, the non-uniqueness is exploited by adding extra axioms to the underlying set theory, while keeping the definition of the real number system within the set theory fixed. This suggests that ZFC is not the appropriate underlying set theory for the hyperreal number system. Set theory might have taken a different direction if it had been developed with the hyperreal line in mind. What is needed is an underlying set theory which proves the unique existence of the hyperreal number system, with the possibility of exploiting the absolute non-uniqueness by adding extra axioms in the same manner

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as is done for the real number system in ZFC. This underlying set theory should have the power set operation to insure the unique existence of the real number system, and another operation which insures the unique existence of the pair consisting of the real and hyperreal number systems. 12.

HYPERREAL SET THEORIES

We have concentrated on the hyperreal line in this article, but we shall now shift to the broader perspective of a 'nonstandard universe' where the hyperreal line appears at the first level. Several set theories have been proposed as foundations for a nonstandard universe. All such theories have a Transfer Principle which guarantees that the hyperreal line is an elementary extension of the full structure on the real line. In this broader perspective, the hyperreal line is not necessarily an ultrapower of the real line, but by the results of (Keisler, 1963) it must be a limit ultrapower. Two approaches, which we shall discuss briefly here, are currently used in the literature. The first approach is now called the superstructure approach. It is related to Robinson's original formulation in (Robinson, 1966) using the theory of types, and is due to Robinson and Zakon (1969). It is often presented by constructing a model within ZFC, but we shall formulate it axiomatically. We shall give this theory the name RZ. We first motivate the theory by describing its intended interpretation. Given a set X, we inductively define

Vo(X) = X,

VW+ 1 (X) = X U Vn(X),

V(X) =-UnVW(X).

V(X) is called the superstructure over X. The intended models of RZ are structures of the form

Mvx), v(Y), *) where X and Y are nonempty sets and * 'V(X) --> V(Y) is an embedding which preserves first-order formulas in the vocabulary {=, E ) with only bounded quantifiers (Vu E v), (3u c v). (To avoid unintended e relationships, X must be chosen so that 0 e X and each x E X is disjoint from V(X), and similarly for Y). The sets in V(X) are called standard sets, the sets in V(Y) are called external sets, and the sets in V(Y) which are elements of *A for some A (- V(X) are called internal sets. The language of RZ is a two-sorted predicate logic with equality, unary predicate symbols X in the first (standard) sort and Y in the second

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(external) sort, a binary relation symbol E for each sort, and a function symbol * whose type is a mapping from the first sort into the second sort. Let Z denote Zermelo set theory with choice and urelements (individuals) but without the axiom of infinity. The axioms of RZ will consist of one copy of the theory Z for each sort with X and Y the sets of urelements, a scheme which says that * is an elementary embedding with respect to bounded quantifier formulas, and an axiom stating that X has a countable subset N and * maps N properly into *N. Both the real and hyperreal number systems exist uniquely relative to the theory RZ. In RZ, one can prove that X contains a copy of N (unique up to isomorphism). Using N and the power set operation, one can then prove in the usual way that there is a unique (up to isomorphism) complete ordered field R at level 1 of the first sort. The hyperreal number system is then characterized uniquely up to isomorphism as the structure (*R, *S : S E 9;) in the second sort. In RZ, the property 'B is internal' is definable by the formula (3A) B E *A. The Saturation Principle is expressible in RZ as follows: For every internal set C, every countable chain B of internal subsets of C has a nonempty intersection. Other properties, such as saturation for higher cardinals, are also expressible in RZ and thus can be added as extra axioms if necessary. Using the ultrapower construction, ZFC proves that RZ is consistent and in fact every superstructure (V(X), E) with an infinite X can be expanded to a transitive model of RZ. Therefore any statement about the standard superstructure which can be proved in RZ can also be proved in ZFC and hence is a theorem according to the usual rules of mathematics. The theory RZ would become inconsistent if the replacement scheme were added (upgrading the Zermelo axioms to the Zermelo-Fraenkel axioms). Hint. Show that every standard set *A has finite rank, because otherwise there would be an infinite decreasing E - sequence below *A.

On the other hand, all of classical mathematics can be carried out in each of the two superstructures in RZ, Moreover, the interaction between the standard, internal, and external sets and the availability of the Zermelo axioms for the external as well as the standard sets makes RZ quite powerful. A second approach is Nelson's Internal Set Theory (IST), introduced in (Nelson, 1977). IST has the equality and E symbols and one additional

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unary predicate symbol st(.) for standardness.The axioms of IST are the axioms of ZFC, the Internalization scheme which is a weak form of Saturation, the Standardization scheme which says that the restriction of any formula to a standard set defines a standard set, and the Transfer scheme which says that the standard sets form an elementary submodel of the universe with respect to the vocabulary {=, E } of ZFC. Compared to RZ, IST retains the replacement scheme of ZFC, but gives up the external sets, leaving only the internal and standard sets. Since IST is an extension of ZFC and uses the st(.) predicate instead of the * function, mathematics in IST looks more like traditional mathematics than mathematics in RZ does. For this reason, it is easier for a classical mathematician to read work in IST than in RZ. However, because the external sets are missing, developments such as the Loeb measure construction and hyperfinite descriptive set theory cannot be carried out in their full generality in IST. In IST, the usual construction of the real line IR gives us the hyperreal line instead, and the restriction of the hyperreal line to the standard predicate gives us the real line. Thus the real and hyperreal lines exist uniquely relative to IST. Using a limit ultrapower construction, Nelson (1977) proved that IST is a conservative extension of ZFC. Therefore, as in the case of RZ, any sentence in the language of ZFC (without the st(.) predicate) which is provable in IST is already provable in ZFC and hence is a theorem according to the usual rules of mathematics. The conservative extension results are often misinterpreted as saying that anything that can be proved with Robinson's analysis can be proved without it. This issue is taken up in (Henson et al., 1984; Henson and Keisler, 1986). As explained, for example, in (Simpson, 1984), almost all of classical mathematics uses only a small part of ZFC, at most second order arithmetic with I-lHcomprehension. In (Henson and Keisler, 1986) it is shown that Robinson's analysis uses principles beyond 1-/i comprehension in a natural way, thus bringing more of ZFC within the reach of our intuition. 13.

DO WE LIVE IN A HYPERFINITE UNIVERSE?

Our intuitive concept of a geometric line is based upon a finite amount of experience with a line in physical space. From this finite experience, we have no way to determine its microscopic structure. For example,

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we cannot tell whether it is finite or infinite, whether it has the Archimedean property, or whether or not it is discretely ordered. Without direct physical evidence, we must fall back on less direct criteria for choosing a mathematical model for the geometric line. One can take a Platonistic view that the geometric line exists, look for properties which the geometric line should have, and represent the geometric line by a mathematical object which has these properties. Alternatively, one can take a pragmatic approach and look for mathematical lines which are useful in explaining or modeling natural phenomena, or in the discovery of mathematical results. One Platonistic view is that the geometric line should be as rich as possible, in some sense containing all possible points. With this criterion, the geometric line should be a fully saturated hyperreal line of large cardinality. Taking this to the extreme, the geometric line should be a hyperreal line which is a fully saturated model whose universe is a proper class. A similar but more convenient alternative is to take the geometric line to be a hyperreal line which is fully saturated model whose size is an uncountable inaccessible cardinal. Another Platonistic view is that the geometric line should be rich but should also be as much as possible like the large finite lines which we know from experience. This leads to the hyperfinite grid. On the hyperfinite grid, Zeno's Paradox is resolved as follows. We can get from 0 to 1 in H steps by taking one step of length 1/H every 1/H seconds, always staying in the hyperfinite grid H. Along the way, we will pass through all the points 1/2, 3/4, 7/8, and so on, since they all belong to the set H. Of course, we will overshoot irrational points such as "ýF2/2, but there will be a time at which we pass from below 4-2/2 to above ý52/2 with one step of length 1/H. A pragmatic argument for the hyperreal line is that it provides a useful source of models for many natural phenomena. There is an extensive literature in mathematical economics (see Rashid, 1987) and physics (see Albeverio et al., 1986) using the hyperreal line. In microeconomics, one studies the behavior of economies with a large number of individually small agents. The large finite economy is represented by a mathematically simpler infinite economy. Large finite economies are sometimes modeled by economies with a continuum of agents, but since the original economy is finite, a hyperfinite set of infinitesimal agents provides a better model than a continuum of agents. A similar approach is useful in physics, where, for example, a large finite set of small

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particles can be modeled by a hyperfinite set of infinitesimal particles. In some Fases, as in the work of Arkeryd (1984) on the Boltzmann equation and the work of Cutland (1986) in control theory, the real line is not rich enough to provide a mathematical representation of a physical phenomenon, and the richer hyperreal line comes to the rescue. The hyperreal line is also helpful in representing phenomena with two scales of measurement where one scale is very large compared to the other. For example, Reeb and his students (see F. and M. Diener (1988) or van den Berg (1987)) have studied singular perturbations by looking through an infinitesimal microscope to classify hyperreal solutions whose trajectories have infinitely fast and slow parts. In physics, the evidence for the existence of an object such as a quark is indirect, and often the only evidence is that the object makes it easier to mathematically represent an observed phenomenon. The hyperreal line makes it easier to mathematically represent natural phenomena, and this may be taken as evidence that the hyperreal line exists in some sense. A second pragmatic argument for the hyperreal line is that it is helpful in the process of mathematical discovery. There are many examples where it has either suggested a fruitful new notion, been used to prove a new result, or been used to give a clearer proof of an old result. The following strategy, sometimes called the lifting method, has been used to prove results which are formulated on the ordinary real line. Step 1. Lift the given 'real' objects up to internal approximations on the hyperfinite grid. Step 2. Make a series of hyperfinite computations to construct some new internal object on the hyperfinite grid. Step 3. Come back down to the real line by taking standard parts of the results of the computations. The hyperfinite computations in Step 2 will typically replace more problematic infinite computations on the real line. Usually, the hard work is in Step 3, where one must show that the appropriate standard parts exist. As a typical example, various existence theorems for stochastic differential equations have been solved by lifting up to the hyperreal line, easily solving the corresponding stochastic difference equation, and then taking standard parts to obtain a solution of the original stochastic differential equation. To date, the lifting strategy has been fully exploited in two areas, probability theory and Banach spaces (see (Albeverio et al., 1986;

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Cutland, 1988; Keisler, 1984) for a variety of applications of this strategy). However, there appear to be possibilities for its use in practically all areas of mathematics. We have stated that the hyperreal line can be used to give a clearer proof of a result. One reason for this is that hyperreal proofs seem to be more 'constructive' than classical proofs. For example, the solution of a stochastic differential equation given by the hyperreal proof is obtained by solving a hyperfinite difference equation by a simple induction and taking the standard part. The hyperreal proof is not constructive in the usual sense, because in ZFC the axiom of choice is needed even to get the existence of a hyperreal line. What often happens is that a proof within RZ or IST of a statement of the form 3xB(x) will produce an x which is definable from H, where H is an arbitrary infinite hypernatural number. Thus instead of a pure existence proof, one obtains an explicit solution except for the dependence on H. The extra information one gets from the explicit construction of the solution from H makes the proof easier to understand and may lead to additional results. Where do we go from here? At the present time, the hyperreal number system is regarded as somewhat of a novelty. But because of its broad potential, it may eventually become a part of the basic toolkit of mathematicians. This process will probably take a very long time, perhaps 50 to 100 years. The current high degree of specialization in mathematics serves to inhibit the process, since few established mathematicians are willing to take the time to learn both mathematical logic and an area of application. However, in the long term, applications of mathematical logic to computer science, as well as applications of the hyperreal numbers, should result in future generations of mathematicians who are better trained in logic, and therefore more able to take advantage of the hyperreal line when the opportunity arises. University of Wisconsin, Madison, U.S.A. REFERENCES

Albeverio, S., Fenstad, F. E., Hoegh-Krohn, R. and Lindstrom, T.: 1986, Nonstandard Methods in StochasticAnalysis and MathematicalPhysics, Academic Press, New York. Anderson, R. M.: 1976, 'A Non-standard Representation of Brownian Motion and It6 Integration', Israel J. Math., 86, 85-97.

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Arkeryd, L.: 1984, 'Loeb Solutions of the Boltzmann Equation', Arch. Rational Mech. Anal., 86, 85-97. Barwise, J.: 1975, Admissible Sets and Structures, Springer-Verlag. van den Berg, I. P.: 1987, NonstandardAsymptotic Analysis, Springer Lecture Notes in Math. 1249. Chang, C. C. and Keisler, H. J.: 1990, Model Theory, North-Holland (Third Edition). Conway, J. H.: 1976, On Numbers and Games, Academic Press. Cutland, N. J.: 1986, 'Infinitesimal Methods in Control Theory, Deterministic and Stochastic', Acta Appl. Math., 5, 105-136. Cutland, N. J.: 1988, Nonstandard Analysis and its Applications, London Math. Soc. Diener, F. and Diener, M.: 1988, 'Some Asymptotic Results in Ordinary Differential Equations', in N. J. Cutland (Ed.), NonstandardAnalysis and its Applications, pp. 282-297, London Math. Society. Henson, C. W.: 1979, 'Analytic Sets, Baire Sets, and the Standard Part Map', Canadian J. Math., 31, 663-672. Henson, C. W.: 1979, 'Unbounded Loeb Measures', Proc.Amer. Math. Soc., 74, 143-150. Henson, C. W., Kaufmann, M. and Keisler, H. J.: 1984, 'The Strength of Nonstandard Methods in Arithmetic', J. Symb. Logic, 49, 1039-1058. Henson, C. W. and Keisler, H. J.: 1986, 'On the Strength of Nonstandard Analysis', J. Symb. Logic, 51, 377-386. Hurd, A. E. and Loeb, P. A.: 1985, Introduction to NonstandardReal Analysis, Academic Press. Jin, R.: 1992, 'Cuts in Hyperfinite Time Lines', J. Symb. Logic, 57, 522-527. Jin, R. and Keisler, H. J.: 1993, 'Game Sentences and Ultrapowers', Ann. Pure and Apple. Logic, 60, 261-274. Kamo, S.: 1981, 'Nonstandard Natural Number Systems and Nonstandard Models', J. Symb. Logic, 46, 365-376. Keisler, H. J.: 1964, 'Good Ideals in Fields of Sets', Ann. Math., 79, 338-359. Keisler, H. J.: 1963, 'Limit Ultrapowers', Trans. Amer. Math. Soc., 107, 383-408. Keisler, H. J.: 1986, Elementary Calculus: An Infinitesimal Approach, Second Edition, PWS Publishers. Keisler, H. J.: 1984, 'An Infinitesimal Approach to Stochastic Analysis', Memoirs Amer. Math. Soc., 297. Keisler, H. J., Kunen, K., Leth, S. and Miller, A.: 1989, 'Descriptive Set Theory over Hyperfinite Sets', J. Symb. Logic, 54, 1167-1180. Keisler, H. J. and Leth, S.: 1991, 'Meager Sets in the Hyperfinite Time Line', J. Symb. Logic, 56, 71-102. Kunen, K. and Miller, A.: 1983, 'Borel and Projective Sets from the Point of View of Compact Sets', Math. Proc. Camb. Phil. Soc., 94, 399-409. Laugwitz, A. H. and Schmieden, C.: 1958, 'Eine Erweiterung der Infinitesimalrechnung', Math. Z., 69, 1-39. Lightstone, A. H. and Robinson, A.: 1975, Non-Archimedean Fields and Asymptotic Analysis, North-Holland. Loeb, P. A.: 1975, 'Conversion from Nonstandard to Standard Measure Spaces and Applications in Probability Theory', Trans. Amer. Math. Soc., 211, 113-122. Log, J.: 1955, 'Quelques Remarques, Th~or~mes et Problkmes sur les Classes D~finissables

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d'Alg~bres', in Mathematical Interpretation of Formal Systems, pp. 98-113, NorthHolland. Luxemburg, W. A. J.: 1962, Non -standardAnalysis, mimeographed Notes, Cal. Inst. Tech., Pasadena, CA. Luxemburg, W. A. J.: 1962, 'Two Applications of the Method of Construction by Ultrapowers to Analysis', Bull. Amer. Math. Soc., 68, 416-419. Morley, M. and Vaught, R.: 1962, 'Komogeneous Universal Models', Math. Scand., 11, 37-57. Nelson, E.: 1977, 'Internal Set Theory; A New Approach to Nonstandard Analysis', Bull. Amer. Math. Soc., 83, 1165-1198. Pincus, D.: 1974, 'The Strength of the Hahn-Banach Theorem', in A. E. Hurd and P. A. Loeb (Eds.), Victoria Symposium on NonstandardAnalysis, pp. 203-248, Springer Lecture Notes 369. Rashid, S.: 1987, Economies With Many Agents: A NonstandardApproach, John Hopkins. Robert, A.: 1988, NonstandardAnalysis, Wiley. Robinson, A.: 1966, NonstandardAnalysis, North-Holland. Robinson, A. and Zakon, E.: 1969, 'A Set-theoretical Characterization of Enlargements', in W. A. J. Luxemburg (Ed.), Applications of Model Theory to Algebra, Analysis, and Probability, pp. 109-122, Holt, Rinehart, Winston. Simpson, S.: 1984, 'Which Set Existence Axioms are Needed to Prove the CauchyPeano Theorem for Ordinary Differential Equations?', J. Symb. Logic, 49, 783-802. Spector, M.: 1988, 'Ultrapowers Without the Axiom of Choice', J. Symb. Logic, 53, 1208-1219. Stroyan, K. D. and Bayod, J. M.: 1986, Foundationsof Infinitesimal Stochastic Analysis, North-Holland. Tarski, A.: 1920, 'Une Contribution ý la thdorie de la mesure', Fund. Math., 15, 42-50. Tarski, A. and Mckinsey, J. C. C.: 1948, A Decision Method for Elementary Algebra and Geometry, 2nd ed., Berkeley, Los Angeles. Zakon, E.: 1969, 'Remarks on the Nonstandard Real Axis', in W. A. J. Luxemburg (Ed.), Applicatons of Model Theory to Algebra, Analysis, and Probability,pp. 195-227, Holt, Rinehart, Winston.

PHILIP EHRLICH

ALL NUMBERS GREAT AND SMALL'

INTRODUCTION

Starting with the ordered set Q of rational numbers, Dedekind defined a real number to be an ordered pair (L, R), where L < R, L U R = Q and L, R # 0. von Neumann later identified each ordinal with the set of all ordinals previously created, so that, 0 = 0, 1 = {0} .... ,o = {0, 1.... }, and so on. More recently, Conway [5, 6] discovered that these two methods can be subsumed under a more general construction which leads to an ordered class of numbers embracing the reals and the ordinals as well as many less familiar numbers including -o, o/2, 1/(o, 4-o and (o - it to name only a few. He further showed that the arithmetic of the reals may be extended to the entire class yielding a real-closed ordered field. Beginning with the empty set of numbers, Conway [6, pp. 4-5] informally constructs his real-closed ordered field by means of the following conventions and recursive clauses. CONSTRUCTION. If L and R are any two sets of numbers and no member of L is > any member of R, then there is a number {LIR}. All numbers are constructed in this way. CONVENTION. If x = {LIR}, we write xL for the typical member of L, and x' for the typical member of R. For x itself we write {x'IxI}, and x = {a, b, c, . . . Ie, f, g .... } is always understood to mean that x = {LIR}, where a, b, c, .... are the typical members of L, and d, e,f, ...the typical members of R. DEFINITIONS

DEFINITION of x > y, x < y. x > y iff (no xR < y and x < no yL), and x < y iffy Žx. 239 P. Ehrlich (ed.), Real Numbers, Generalizations of the Reals, and Theories of Continua, 239-258. © 1994 Kluwer Academic Publishers. Printed in the Netherlands.

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DEFINITION of x = y, x > y, x < y. Sx=y iff (x Žy andy Ž x). x>y iff (x Žy and not [y Ž x]). x < y iff y > x. DEFINITION of x + y. x + y = {xL + y, x +yLxR + y, x +y. DEFINITION of -x. -x = {-xIxL}. DEFINITION of xy. xy = {xLy + xyl - xLyL, xRy + xyR - xRyR xLy + xyR - xyR, XRy + xyL - xRyL}.

This particular real-closed field, which Conway calls No, is remarkably inclusive. Indeed, subject to the proviso that numbers be individually definable in terms of sets of von Neumann-Bernays-Gbdel set theory with Global Choice, henceforth NBG (cf. [14]), it may be said to contain 'All Numbers Great and Small' [6, p. 3] or all finite, infinite, and infinitesimal numbers. From a (first-order) logical standpoint, this may be made precise by saying that No is (up to isomorphism) the unique absolutely saturated model for the theory of real-closed orderedfields, absolutely = On being the cardinal of all proper classes in NBG [10]. However, in addition to its distinguished structure as a maximal realclosed ordered field, No has a rich hierarchical structure which emerges from the recursive clauses in terms of which it is defined and which lies at the heart of its logical basis. This additional structure, which may be described as a simplicity hierarchy, is intimately dependent on No's (implicit) structure as a lexicographically ordered binary tree and arises from the fact that the sums and products of any two elements of the tree are the simplest possible elements of the tree consistent with the theory of real-closed ordered fields, it being understood that x is simpler than y just in case x is a predecessor of y in the tree. Simplicity is essentially a relation of dependency, since x is simpler than y just in case y cannot be defined unless x has already been defined. Since the predecessors of any element are created earlier than the element itself, this insures that the sums and products of any two elements of the tree appear in the tree just as soon as they possibly can.

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In the pages that follow an alternative treatment of Conway's number system will be presented in which the algebraico-tree-theoretic features of the system are brought to the fore. In Section 1 an explicit definition of a number2 is provided, making use of a natural generalization of von Neumann's definition of an ordinal. Whereas an ordinal, for von Neumann, emerges as the set of all of its predecessors in the ('long' though rather trivial) binary tree (On, E ) of all ordinals, each number x in our system is an ordered pair (L., R.), where L, (resp. R,) is the set of all the predecessors of x to the left of (resp. to the right of) x, in the complete binary tree (No, <s) of all numbers. In Section 2, with the tree of numbers before us, we proceed to impose the rule of order, recover Conway's theory, and in the process see that if our numbers obey the rule, then they may indeed set the (settheoretic) standard for All Numbers Great and Small! This being the case, the ordered number tree is transformed into an absolutely saturated real-closed ordered field a la Conway, albeit now with a tree-theoretic definition for the symbol '{LIRI'. Following this, in Section 3, we introduce a representation-independent characterization of a lexicographically ordered binary tree, see that the rule of order is just the lexicographical order applied to the tree of numbers, and establish characterization theorems which distinguish the tree of all numbers great and small from the remaining ordered trees. The latter theorems are then employed in Section 4 to help establish characterization theorems which, in effect, assert that (No, +, -, <, <s) has all of the algebraico-tree-theoretic structure we attributed to it earlier. Finally, in Section 5, we see that insofar as each of our numbers has its own unique properties, the system of numbers considered as a whole has no nontrivial automorphism. Of course, being an absolutely saturated real-closed ordered field, (No, +, ., <) has a plethora of automorphisms. However, with the exception of the identity automorphism, these do not preserve simplicity. This may be interpreted as showing that if you want to know which number one of our numbers is, you should not ignore from whence the number came. 1.

THE NUMBER TREE

von Neumann defines an ordinal as a transitive set that is well-ordered by the membership relation. The following definition extends the kernel of this idea to numbers more generally.

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DEFINITION 1.1. A number x is an ordered pair (Lx, R.) of disjoint sets such that: (i) if y E Lx U Rx, then y is an ordered pair (LY, Ry) of disjoint sets where Ly C Lx and Ry C R.; (ii) there is a well-ordering <, of Lx U R, such that for each y = (Ly, Ry) E Lx U Rx, Ly U Ry = {Z E Lx U Rx : z <x y}. 3 THEOREM 1.1. (i) (0, 0) is a number; (ii) If x is a number, then (Lx U {x}, Rx) and (Lx, {x} U Rx) are numbers; (iii) if (xa).

XiB.

Xv

Xv

v

impossible, and so L n R = 6. Now suppose y E L U R. Since for some < 3,y E L U Rx. where x, is a number, y is a disjoint ordered pair of sets. Moreover, since Ly C L, C L and Ry C R_ C R, we have Ly C L and Ry C R. To show that x = (L, R) is a number it only remains to note that since for each gI < 3, <x is a well-ordering of Lx U Rx such that for each E Lx. U R., LY U RY = {z E L U Rx :z<x. y}, and since for all gX< v < I0, <XV is an extension of <x•, it Xjollows that <x = U•,, <x, is a well-ordering of L U R such that for each y E L U R, LY U RY = {z E L U R : z <X y}. Turning now to (iv), let x E No and let (x ),a,
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DEFINITION 1.2. No = {x : x is a number}. DEFINITION 1.3. Let x, y e No. We will say x <s y (x is simpler than y) iff x E LY U Ry. We will now begin the process of showing that if x e No then Lx U Rx is the set of all the predecessors of x in the complete binary tree (No, <s) of numbers. PROPOSITION 1.1. If x ( No and y E Lx, U Rx, then y E No. Proof. The result clearly holds for x = (0, 0). Moreover, if the result holds for x = (Lx, Rx), then it obviously holds for (Lx U {X}, Rx) and (Lx, {x} U Rx); and if the result holds for xa whenever ox < an infinite limit ordinal P3,then it also clearly holds for (U(<,Lxx, Ua
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two immediate successors and every chain of limit length has at most one immediate successor [7, p. 216]. A binary tree is said to be complete iff every member has two immediate successors and every chain of limit length has an immediate successor. The paradigm example of a complete binary tree is the class U•<,{00, 1} partially ordered by inclusion, henceforth (T, <0). THEOREM 1.3. (No, <,) is a complete binary tree. Proof In virtue of Theorem 1.1 and Proposition 1.2 it only remains to show: (i) (L, U {x}, R,) and (Lx, {x} U Rx) are the immediate successors of any x E No; and (ii) (U
R, L,

ifx, e Lx if xa E Rx.

In virtue of Theorem 1.3, the following result is to be expected. THEOREM 1.4. (i) Every number has a (unique) genetic code; (ii) distinct numbers have distinct genetic codes; (iii) every sequence of elements of {L, R} is the genetic code of some number. Proof Obviously, (i) is the case. Suppose then that x, y E No and let (x.),<, and (y),,,•p be their respective genealogies. Since (No, <,) is a binary tree, if x # y and (x,),a
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g(x) # g(y). On the other hand, if 13= p, there is a g. < P such that xa = y, for all c < V.and x, # y,. Moreover, V.is not a limit ordinal; for if it were x,(= y,,) would be the immediate successor of the chain (x,),<,. But if g. = a + 1, x, and y, are the distinct immediate successors of xo, in which case g,(x) # g,(y), which proves (ii). Finally, let s = (s,)a,, be a sequence where P3 E On and s,, E {L, R1 for all a < •. If 13= 0, in which case s = 0, s is the genetic code of (0, 0). And if P3 > 0, suppose that for each a < P3,(sJ),)<6 is the genetic code of some number x, = (Lx,, Rx,). If 13= a + 1, then s is the genetic code of (L., U {x.}, Rx,) (resp. (Lx,, {x.} U RXo)) if s, = R(resp. L); and if P3 is an infinite limit ordinal, s is the genetic code of (Ua<•LX, Ua
ALL NUMBERS GREAT AND SMALL

Our definition of order is motivated by the following intuitive idea: x < y iff x is to the left of y or y is to the right of x or there is some z which is both to the right of x and to the left of y. More formally, we have DEFINITION 2.1. (The Rule of Order). For all x, y y E R. or R, n Ly •0.

xE Ly or

E

No, x < y iff

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THEOREM 2.1. Numbers are totally ordered by the Rule of Order. Proof Let x, y, z E No and let a (resp. b) be the element of greatest tree-rank in (prNo(x) U {X}) n (pruo(y) U {y}) (resp. (prNo(y) U {y}) n (prNo(z) U {z}). (Connectivity): Suppose x # y. Either (i) x E PrNo(y) or (ii) y e prNo(x) or (iii) not [(i) or (ii)]. For each of (i) and (ii), clearly x < y or y < x. Moreover, by (iii), the hypothesis on a and the fact that x y, eithera E Rx n Lyora r Ry n Lx, and sox, > and = are defined by the same definitions as for numbers and the class of games is recursively generated by the principle: If L and R are any two sets of games, then there is a game {LIR}. All games are constructed in this way. In our approach, on the other hand, the limitations of formalization

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in NBG never arise, there is no need to first introduce the class of games (but see Appendix III), and, as we will now see, there are no salient features of Conway's number system which are lost in the transition. DEFINITION 2.2. Let (L, R) be a pair of subsets of No for which L any member of R (i.e. L < R), there is a number {LIR). All numbers can be characterized in this way. Proof. LetA= {xe No:L<{x} < R)}, and for each E On let Op = {y = No : pNo(Y) < 0} and No = Op+, - Op. Since IL U RI < On, there is a least member of On, say, 't, such that L U R C 0_. Furthermore, there is a greatest ordinal < t, say, a, such that for all a < a, A n Na = 0. Moreover, for each such ax there is a unique partition (La, Ra) of Na such that L is cofinal with La and R is coinitial with Ra; and finally, we have: (*) L is cofinal with U, 0 and for each ax < a, Lza is a cofinal subset of U<,,aLp and R2,, is a coinitial subset of UpaRp. Plainly then, (**) holds if a is an infinite limit ordinal. On the other hand, if Y= a + 1, then clearly za E Na, and by Definition 2.1 Lza < {za}< Ra. Moreover, if we now suppose there is another number, say, z, where La < {z) < Rz and z r Na, by Definition 2.1 there would be a w lying between z and za (and hence between Lza, and Rza) where w E 0 a,. But since (Up<,Lp, U<,,Rp) is a partition of Oa where UpaLp < Upa,Rp, this is precluded by the induction hypothesis; and so Lz < {z} fn Na < Ra iffz = za, which implies (**). With (**) at hand we may now appeal to (*) and the kind of argument just employed for the case of a = a + I to see that A n Na = {zo}. Now suppose x r A - {zJ. Then either z, < x or x < z,. But

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since PNo(ZG) < PNo(X) and for no c both pNo(C) < PNo(Z,) and z6 < c < x (resp. x < c < z,) hold, it follows from Definition 2.1 that z, c Lx (resp. R.) and so z6 = {LIR}, thereby proving the first portion of the theorem. To establish the second portion one need only appeal to the following simple observation. THEOREM 2.3. If x is number, then x = {LIRx}. Proof. If x # z = {LxIRj}, then since Lx < {x} < R,, it follows that z <s x and so z - L, U Rx, which is obviously impossible.

THEOREM 2.4. For all x = {xlix)} and y = {ymLyR} in No, (i) x= y iff each xz < y < each x' and each yL < x < each yR; (ii) x y iff x < each yR and each x' < y; (iii) x < y iff x some yL or some xR < y. Proof If x = y, then obviously each XL < y < each xR and each yL < x < each yR; and if the latter two conditions hold, then either x = y or (x <s y and y <s x). But since <s is a strict partial order x <s y implies not (y <s x), and so x = y, thereby proving (i). Now suppose x • y. Then x • y < each yR and each xL <x • y; and so x < each yR and each xl' < y. Moreover, if the latter two conditions hold and x < y does not hold, it follows that each y' < y < x < each yR and each xL < y < x < each xR. But then by (i), x = y which is impossible, and so (ii) is the case. Finally, to establish (iii) we need only appeal to (i) and (ii). THEOREM 2.5. The ordered class No with +, - and • defined by recursion A la Conway (albeit now with our definition for the symbol '{ I}') is isomorphic to Conway's orderedfield. Proof Suppose F is real-closed. Then F is isomorphic to Conway's ordered field iff for all subsets X and Y of F where X < Y, there is a z E F such that X < {z I < Y ([9]; see also [ 1]). Thus, in virtue of Theorem 2.2, it suffices to show that (No, +, ., <, 0, 1) is a real-closed ordered field, where 0 = (0, 0) and 1 = ({0}, 0). But since (with Theorems 2.2-2.4) we have recovered the essential features of Conway's system of All Numbers Great and Small, we can now employ Conway's proof (almost symbol for symbol) to establish that (No, +, -, <) is a real-closed ordered field. (For the induction principles required to carry out Conway's proof, see Appendix II).

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ALL NUMBERS GREAT AND SMALL 3.

THE STRUCTURE OF (No, <, <s)

The lexicographic ordering for binary trees is usually defined for initial subtrees of (5, <0) or notational variants thereof. In the case of ,
R.

Unlike these definitions, however, the following characterization of a lexicographically ordered binary tree is free of the idiosyncracies of any particular representation. Moreover, the appelation is certainly appropriate since the definition provides an answer to the question: What must be true of an ordered binary tree (T, <,
=

({f(y) : y e LT(x)1, (f(y) : y e Rrx)}),

is the unique monomorphism of ordered trees from T into No for which PNo(f(x)) = PA(x), for all x e T. Clearly then, to substantiate our contention about Definition 3.1 we need only establish THEOREM 3.1. (i) (No, <, <s) is a lexicographicallyordered complete binary tree. (ii) g : No -ý (35 is an isomorphism of ordered trees. Proof (i) is evident; and by Theorem 1.5 (ii), g is an isomorphism of trees. Moreover, by Theorems 1.5 (iii) and 2.1 we have (0)

x

<

y iff g(x)

E Lg(y)

or g(y) e Rg(x)

or

Rg(x)

n Lg(y)

#

0.

But, by Definition 1.5, g(x) e Lg(y) iff (i): g,(x) = g,(y) for all x < 8 = pj(g(x)) and g8(y) = R; and g(y) e Rg(X)iff (ii): g,(x) = g,,(y) for all cx < 8 = pQ$(g(y)) and g8(x) = L; and Rg(x) n Lg(y) ;- 0 iff (iii): there is a g(z) e W such that g,(x) = ga,(z) = g,(y) for all cx< 8 = p9(g(z)) and g8(x) = L and g8(y) = R. Therefore, since the disjunction of (i)-(iii) is clearly equivalent to the condition which defines the lexicographic

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ordering for ((M, <,), we have x < y iff g(x)
g(y), for all x, y E

Our next two theorems shed further light on the structure of (No, <, <s). THEOREM 3.2. Let (T, <, 0. If a = a + 1 for some (, then there is a z = (LT(z), RT(z)) = (L,, Rz) which is the immediate predecessor of x in T,,i. Moreover, prT•+l(x) = prNo(z) U {z}. But then by the hypotheses on x and <, either x = (Lz, {z} U Rz) or x = (Lz U {z}, Rz), which proves Ta,.l = No,+,. On the other hand, if x • Y + 1 for some ay, then x is the immediate successor in TatI of some chain (x,),
ALL NUMBERS GREAT AND SMALL 4.

251

A CHARACTERIZATION THEOREM FOR (No, +, -, <, 0, 1, <s)

A (class) model Wffor a (first-order) theory T in a language L (of power < On) is said to be absolutely saturated iff for every subset X of the universe of W.the expanded model (%., a)acx realizes every type X.(v) of the expanded language L U {Ca : a E X} which is consistent with the theory of (%., a)aex [101. Roughly speaking, absolutely saturated models are the maximal models of first-order theories having infinite models and, as we already noted, Conway's number system is (up to isomorphism) the unique absolutely saturated real-closed ordered field. To this we may now add. THEOREM 4.1. (No, +, ., <, 0, 1) is (up to isomorphism) the unique absolutely saturated real-closed orderedfield. As our introductory remarks suggest, however, (No, +,0, <, 0, 1) is no 'mere' absolutely saturated real-closed ordered field, and (No, +, -, <, 0, 1, <s) is not simply such a model supplemented with a lexicographically ordered binary tree structure. To establish the characterization theorem for (No, +, ., <, 0, 1, <s) which we alluded to in the introduction we will rely in part on two crucial insights implicit in the following remarks offered by Conway to motivate his definitions of +, -, - and <. Now it is our intention that each new number x shall be not > any XR,and not < any XL,

and that >, +, -, • shall have the usual properties. So if (say) y > some X8 we could not have x > y, for then x > x". Similarly we could not allow x > y if x < some yL. So we define x > y in all other circumstances. Again, we cannot allow x + y to be < XL + y or x + yL, or > XR + y or x + yR, justifying our definition of x + y, and we cannot allow -x < -xYor -x > -XL, justifying the definition of -x. For xy we observe that (for instance)

(x - xL) and (y - yL) are $ 0, so that we must ensure that the same holds of their product, whence xy $ XLy + xyL - XLyL, and the other entries in the product definition are similar. So that we may say that the definitions of the various operations and relations are just the simplest possible definitions which are consistent with their intended properties ([5], pp. 2-3; see also [6], pp. 5-6).

THEOREM 4.2. Let 2 be the theory of real-closed orderedfields and let T = (T, +,, ",, 0 ', 1 , <
252

PHILIP EHRLICH

Proof. Since (T,
any

XL, XR, yL

(*)

and

yR

we have

XL < x < x'

and

yL < y
But since No is an ordered groupoid it follows that xL +y<x+y<xR+y

x+yL <x+y<x+yR

and

and hence that (**)

{xL+y,X+yL} < {x+y} < {xI+y,x+yR}.

Moreover, since No is an ordered group, it follows from (*) that each of the differences x - xL, y - yL, xR _ x, yR _ y

is positive, and consequently, so is each of the products (x - xL)(y _ yL), (XR _ x)(yR

(X _ xL)()ý _ Y), (XR _ x)(y

y),

-yL),

since No is an ordered field. Therefore, by applying the routine algebra of ordered fields to each of these products one obtains (***)

{xLy + xyL - XLyL, xRy + xyR - XRyR} < {Xy} < {xLy + xyR _ XLyR, XRy + XyL _ XRyL}.

Thus, since No is a model of

Z3,

x + y and xy must 'lie between' the

(possibly empty) sets of elements of No specified in (**) and (***), respectively. But since x + y and xy are, by definition, the simplest elements of No 'lying between' these sets, to complete one direction of the proof we need only invoke the real-closed nature of No to insure the satisfaction of the consistency requirement. To establish the converse, one employs arguments like those used to obtain (**) and (***), together with the hypotheses on x +, y and x ", y, to obtain X +T Y = {XL +TY, X +T yLIxR +T Y, X +T yR}

and the analogous result for x

*TY.

One now proves by means of a

trivial induction (see Appendix II), that x +T Y = x + y. The induction

ALL NUMBERS GREAT AND SMALL

253

hypothesis is that xL +T y = xL + y, x +T yL = x + y', etc., and the basis step holds vacuously since it rests on answers to questions about members of the empty set. Following this, a similar inductive argument shows that x "TY = xy. Since the absolutely saturated real-closed ordered fields are precisely the real-closed fields that are ilo,-orderings, it should now be clear that in addition to Theorem 4.2 we also have THEOREM 4.3. Let Z3 be the theory of real-closed orderedfields and let IF = (T, +T, *T' OTI T,
INDIVIDUATING

NUMBERS

Unlike the ordered field of reals which has no nontrivial automorphism, No considered merely as an ordered field has many. Indeed, since No is an llon-ordering it obviously has On distinct automorphisms of its ordered field structure. However, as we already suggested and will now show, all but one of these automorphisms fail to take into account from whence the numbers came. THEOREM 5.1. (No, +, ., <, 1, 0, <s) has no nontrivial automorphism. It suffices to show that (No, <, <s) has no nontrivial automorphisms. Let f by an automorphism of (No, <, <s). Since (0, 0) is the unique x E No for which prNo(x) = 0 and f is a surjection, f((0, 0)) = (0, 0). Now suppose x e No and f(y) = y for all y e prNo(x). Since f preserves <s and f is a surjection, clearly PNo(X) = pNO(f(x)). If PNo(X) = (x + 1 for some cx, then either f(x) = x orf(x) = x', where x' is the other immediate successor of the immediate predecessor of x. But since f preserves <, it must be x. Moreover, if PNo(X) is a limit ordinal, then f(x) = x since if y e No and prNo(y) = prNo(X), then y = x. Closely related to Theorem 5.1 is

254

PHILIP EHRLICH

THEOREM 5.2. (No, +, ., <) has no nontrivial automorphism f such that for all x E No, prNo(f(x)) = prNo(x). This rests on the simple observation that the preservation of order and sets of predecessors implies the identity map. Even stronger results of this kind can be obtained using THEOREM 5.3 Let f: No -- No be a mapping. The following are equivalent. (i) f is the identity map. (ii) g(f(x)) = g(x), for all x E No. (iii) f(x) = {f(xL)Lf(xR)}, for all x E No. APPENDIX I

Let Definition 1.1' be the definition of a number that results from replacing condition (ii) in Definition 1.1 with (ii)'

{Ly U RY U {y) : y = (Ly, Ry) E Lx U Rx}

is well-ordered by strict inclusion. Definition 1.1' is slightly more in the spirit of von Neumann's definition of an ordinal than is Definition 1.1. There is, however, an alternative (and perhaps simpler) characterization of a von Neumann ordinal which is closer in spirit to Definition 1.1. Namely, A set x is a von Neumann ordinal iff there is a well-ordering <, of (x such that for each y e x, y = {x e u. : x <, y}. The following proof of the equivalence of Definitions 1.1 and 1.1' depends on the notion of set-theoretic rank and hence on the Axiom of Foundation (cf. [13], Ch. II). THEOREM. Definitions 1.1 and 1.1' are equivalent. Proof. (ii) clearly implies (ii)'. Now suppose (ii)' and let (y•)a(• be the enumeration of Lx U Rx defined by the condition: for all g, v < 3, [ } C Lyv U Ryv U {y,}. Since y,• y, whenever yg < y, ifLy U R U {y jg # v, we clearly Lave: (1) for all gt < v < 03, y, E Lyv U Ryv. But since any member of LyV U Ryv must have a lower set-theoretic rank than y, = (Lyv, RY) = { {Ly}, {Lyv, Ry} }, it is evident in virtue of (1) that: (2) for all gt, v < P3,yv e LyL U RyUif V > gI. But in virtue of Definition

ALL NUMBERS GREAT AND SMALL

255

1.1 (i) yE L URwheneveryE L• U UR, and g < P; and so we also have: (3) for all g < P, ifyE Ly U R,., then y = y, for somev
A relation R on a class A is said to be well-founded iff every non-empty subset of A has an R-minimal member and for every x c A, {z E A : zRx} is a set. The following induction principle is a standard theorem in the theory of well-founded relations (cf. [13], p. 64): (Induction on a Well-Founded Relation R): If Vx[Vy(yRx -- 'Dy)

--

Ox], then VxADx. Since (No, <,) is a tree, <s is of course a well-founded relation on No (cf. [13], p. 293) and so we have the following counterpart of the induction principle for numbers employed by Conway ([6], pp. 5, 16, 64): Principleof Inductionfor Numbers for a property P: If for every number x, Px holds whenever Px, holds for every number x' which is simpler then x, then P. holds for every number x. Furthermore, by employing iterated applications of this principle we may also obtain our counterpart of Conway's induction principle for a relation (and/or an operation). Induction Principlefor Numbers for a Relation R: If for all n-tuples (x1..... x.) of numbers R. .... n holds whenever Rxl ... xý holds for every n-tuple (x' ..... x') of numbers where, for every i(1 < i < n)x' <s xi, and for some such i, x' <s xi, then Rx1 . holds for all n-tuples (x1 .....

x.) of numbers.

It is perhaps worth emphasizing that while Conway's induction principles for numbers can be justified along the above lines regardless of which method of number construction is employed, since our numbers are ordered pairs of sets of simpler numbers, the following alternative justification of the first principle is possible ([8], Ch. 13).

256

PHILIP EHRLICH

Proof. Suppose the contrary. That is, suppose there is a number x for which -P, holds, but for every number x, Px holds whenever P., holds for every number x' E L, U R.. Then there is a number x0 E Lý U Rx such that -Pxo, and hence a number x, E L,0 U R.o such that -P, so on. Clearly then, for each n < o we have: x,,, = (Lý,+, Rx+ 1 ) = {Lx+,,}, {Lx,+,, R+,,,} } E (L. or Rx.) E {Lx,, Rx} E {{Lx}, {L,,, Rj}} = (Lx., Rýý) = x.. But since this plainly requires the existence of an infinite sequence of sets Yo, Y .... where ... E y E Yo, it violates the Axiom of Foundation. APPENDIX III (PART 1)

While our construction of the class of numbers does not presuppose the prior introduction of Conway's class of games, inclination and/or wider interest might lead one to adopt such an approach. With this in mind, it is perhaps worth noting that if one avails oneself of this option it becomes possible to generate our number class by means of the following inductive construction that is reminiscent of Conway's. CONSTRUCTION. If L, R is a partition of the chain of all numbers simpler than (L, R) and no xL > (L, R) > no xR whenever x E L U R, then (L, R) is itself a number. All numbers are constructed in this way. This principle may be regarded as inductively extracting the numbers from the universe of a structure (G, >, <,'}, where > is defined b la Conway, and <,' is simply the extension of <s (see Definition 1.3) to the more inclusive class G of games, i.e., the class recursively generated by the principle: If L and R are any two sets of games, then there is a game (L, R). All games are constructed in this way. < may be defined b la Conway or with our Definition 2.1. Alternatively, those who prefer to work directly with the tree (G, <s,) can inductively isolate the numbers via the principle: If L, R is a partition of the chain of all numbers simpler than (L, R) and xL E L and x' E R whenever x E L U R, then (L, R) is itself a number. All numbers are constructed in this way.

ALL NUMBERS GREAT AND SMALL

257

Part 2 In [2, 3 and 9] Alling and/or the author provided an alternative construction of the surreal numbers in which each number is an ordered pair (L, R) where L, R is a canonical partition of all numbers created at earlier levels of recursion. It is also perhaps worth noting that beginning with the structure (G, 5, <,ý), one may inductively extract the class of numbers defined in [2, 3 and 9] by the following construction where <s, is now said to be the created earlier than relation. If L, R is a partition of the set of all numbers created earlier than (L, R) and no member of L is Ž_any member of R, then (L, R) is itself a number. All numbers are constructed in this way. The resulting structure may now be turned into a lexicographically ordered binary tree by asserting that x is simpler than y if Lx < {y} < R, and y # x, where the relation < is defined as in Part 1. Moreover, if one prefers to work with an explicit (as opposed to an inductive) definition of the numbers obtained via this generalization of the Dedekind Cut construction, one may use the following characterization. A number x is a disjoint ordered pair (Lx, Rx) of sets where L, U Rx is the set of all ordered pairs y = (Ly, Ry) for which Ly, Ry is a partition of all the members of Lx U Rx having a lower set-theoretic rank than that of y and for which either Ly U {y} C Lx or RY U {y} C Rx. NOTES During the past few years the author has had the good fortune to be able to discuss

numbers great and small with J. H. Conway on a number of occasions. It is safe to say that were it not for these stimulating conversations the present paper would not have been written. On p. 23 of [61, Conway characterizes the simplest number lying between two sets L and R of numbers, where L < R (i.e., every member of L precedes every member of R) as the earliest created number lying between L and R. This, along with a number of other remarks, led Ailing and the author in [2-41 to identify the 'simpler than' relation between numbers with the 'born earlier than' relation. Regretting that he had not better conveyed his intention, Conway urged the author to identify the simpler than relation

with the predecessor relation in the number tree. Motivated by this suggestion and the conviction that Conway's reduction of the theory of numbers to the theory of games is important, and should therefore be preserved, the author sought to develop a system of

numbers in which each number is itself a game (i.e., an ordered pair of sets of games)

258

PHILIP EHRLICH

and where the chain of dependence associated with the predecessor relation is reflected in a simple and highly visible manner by the set-theoretic relation between the numbers themselves. This naturally led to the generalization of von Neumann's treatment of ordinal numbers presented in Section 1. In an important sense, the approach to numbers presented there may be regarded as an incorporation of the theory of sign-expansions [11, pp. 29-30], [4] into the theory of games. Sign-expansions (= genetic codes) for numbers continue to arise in our approach (see Section 1), though our introduction of them is quite different than Conway's. 2 Those numbers which, following Conway, we simply call numbers have been dubbed SurrealNumbers [12]. Although a more general interpretation is possible, Surreal Numbers may be regarded as the members of ordered (number) fields. Phrases like 'All Numbers Great and Small' may therefore be understood to be referring to such numbers. See Appendix I and Appendix III for alternatives to Definition 1.1.

REFERENCES I. Ailing, N.: 1985, 'Conway's Field of Surreal Numbers', Trans. Am. Math. Soc., 287, 365-386. 2. Ailing, N.: 1987, Foundationsof Analysis over Surreal Number Fields, North-Holland Pub. Co., New York. 3. Ailing, N. and Ehrlich, P.: 1986, 'An Alternative Construction of Conway's Surreal Numbers', C.R. Math. Rep. Acad. Sci. Canada,8, 241-246. 4. Ailing, N. and Ehrlich, P.: 1986, 'An Abstract Characterization of a Full Class of Surreal Numbers', C.R. Math. Rep. Acad. Sci. Canada, 8, 303-308. 5. Conway, J. H.: 1972, All Numbers, Great and Small, The University of Calgary Math. Research Paper No. 149, February. 6. Conway, J. H.: 1976, On Numbers and Games, Academic Press, London. 7. Drake, F.: 1974, Set Theory, An Introduction to Large Cardinals,North-Holland Pub. Co., Amsterdam. 8. Ebbinghaus, H.-D., et al.: 1990, Numbers, Springer-Verlag, New York. 9. Ehrlich, P.: 1988, 'An Alternative Construction of Conway's Ordered Field No', Alg. Univ., 25, 7-16. Errata, Ibid., 233. 10. Ehrlich, R.: 1989, 'Absolutely Saturated Models', Fund. Math., 133, 39-46. 11. Gonshor, H.: 1986, An Introduction to the Theory of Surreal Numbers, Camb. Univ. Press, Cambridge. 12. Knuth, D.: 1974, Surreal Numbers, Addison-Wesley Pub. Co., Reading MA. 13. Levy, A.: 1979, Basic Set Theory, Springer-Verlag, New York. 14. Mendelson, E.: 1979, Introduction to Mathematical Logic (Second Edition), D. van Nostrand Co., New York. 15. Reinhardt, W. N.: 1970, 'Ackermann's Set Theory Equals ZF', Ann. Math. Log., 2, 189-249.

DIETER KLAUA

RATIONAL AND REAL ORDINAL NUMBERS

1.

INTRODUCTION

In consequent extension of Cantor's theory of ordinal numbers 0, 1, 2, o02, o0.2, 0o.2 + 1, o .2 + 2 ....... ... 0c, o + 1,co+ 2 .... .......... non-Archimedean number models for the real axis have been constructed by Sikorski [9] and, independently, Klaua [4, 5]. Reference [9] introduced integral and rational ordinal numbers; references [4, 5] introduced integral, rational, and real ordinal numbers. The purpose of these constructions is to extend the real axis into the transfinite, at the same time as refining it infinitesimally. The domain of ordinal numbers is (reflexively) well-ordered by the not-greater relation := and irreflexively well-ordered by the accompanying smaller relation <. The usual addition and multiplication of ordinal numbers are neither commutative nor absorption-free. Hence they are not suitable for the construction of a transfinite number system of integral, rational, and real ordinal numbers. But, according to Hessenberg, one obtains an addition + and a multiplication • with natural arithmetical properties (in particular with commutativity and absorption-freeness) by adding and multiplying the Cantor normal forms of ordinal numbers like polynomials. The paper is based on these Hessenberg operations +,. since they have the properties needed for the development of a desired transfinite number system, namely, for ordinal numbers cx, P3,y:

S+

+

==13)+ (P +

= a.(o3 + Y) = a.f3 +

3
+ y,

ax

Y,

0 A 3 < y = a.1 < a.y.

On N, the operations +, are the ordinary addition and multiplication of natural numbers. For ordinal numbers o the initial number (o) is the smallest ordinal number oa such that the interval [0, a[ has cardinality X. The interval [0, cow[ is closed under + and -. This fact suggests the possibility of developing a system of integral, rational, and real ordinal numbers over 259 P. Ehrlich(ed.), Real Numbers, Generalizationsof the Reals, and Theories of Continua, 259-276. © 1994 Kluwer Academic Publishers.Printed in the Netherlands.

260

DIETER KLAUA

(0, 3oj, for all initial numbers (o.. For 30 = (o one gets the classical number system Z, Q, R, which is essentially different from the transfinite number systems for the o3, with Y > 0. Therefore we deal only with the case a > 0. Furthermore we often need for our considerations that [0, w0,( is closed under the formation of least upper bounds sup•q cxý of P3-sequences (c4)g 0. Those numbers are for instance the 0o,with isolated index Y > 0 (i.e. a has a predecessor), in particular 0o. One can visualize the theory for (o, where all intervals [0, x[ with cXE [0, 0o,[ are countable, i.e. finite or countably infinite. In the appendix we give, as a modification of ZFC-set theory, an axiom system for the underlying set theory (and for the whole of mathematics): (I) Axiom of urelements; (II) Axiom of extensionality; (III) Axiom schema of separation; (IV) Axiom of pairing ; (V) Axiom of union; (VI) Axiom of power; (VII) Axiom of restriction; (VIII) Axiom of choice; and (IX) Axiom of universe. The elementary axioms I-VIII suffice for the construction of our transfinite number systems. In the given set theory the rank relation can be defined a priori, and by the rank relation one gets (without any additional postulate on the cardinality of the urdomain U), for every set the existence of an equipollent urset U (not necessarily U C U), which is the basis for a superstructure over U used in Nonstandard Analysis. 2.

RATIONAL ORDINAL NUMBERS

Let N, = [0, 3[ w0 be the set of natural ordinal numbers of type 0o. In N, we consider the usual order relations -_s, <, and the Hessenberg operations +, .. By the equivalence relation difference equality =, in No x N• one gets the set Z, = (N0, x N)! =D of integralordinal numbers of type o3, as the set of equivalence classes of =D. By <=,<, +, • from N0 , the order !=, <, addition +, and multiplication • of Z, are defined representative-wise. Then (Z, +, ., :-5) is a totally ordered (commutative) ring. The equivalence relation quotient equality =Q in Z, x (Z"\{0}) yields the set Q,, = (Z, x (74\{0}))/ =Q of rational ordinal numbers of type (o, as the set of equivalence classes of =Q. By :, <, +, • from Zn, the order -5, <, addition +, and multiplication • of Q 0 are defined representative-wise. Then (Q0 , +, ., •-5) is a totally ordered (commutative) field. By isomorphic embedding one has

RATIONAL AND REAL ORDINAL NUMBERS

261

and R, is the common cardinality of these number domains. For the set N = [0, (o[ of natural numbers holds N C N, and we call 7= {x- yjx, y E N I

Z)

Q

=

{x/ylx, y

_AZyO} C Qy

the set of (classical) integralnumbers resp. the set of (classical) rational numbers. Within Q, (and 7Z6, N,) there is an Archimedean axiom of type co,: For every a, b c Qo with a, b > 0 there is an n E N6 with a < b .n. But in the classical sense, Q, (and Z/, N,) is non-Archimedean because, for example, co > I n for all n ( N. According to the classical concept of convergence of real 0)-sequences, we call an co,-sequence a = (a4)ý Žo If such an o: for a exists, it is uniquely determined as the limit limý<,, aý of a. By analogy with the classical case, also co6-fundamental sequences (co,-Cauchy sequences) and co)-nests of intervals are defined in Q,. Finally, a cut in Q, is an ordered pair (A, B) of sets such that (1) (2)

A, B # 0, A U B = Qo, a
and an 0-cut is an ordered pair (A, B of sets satisfying (1), (2), and (3)

For every E with 0 < EE Q, there are a E A, b E B with b -a<.

In the classical rational number field Q not every bounded (0o-)sequence contains a convergent subsequence. This property was first discovered to be valid in the field R of real numbers, where it is known as the BolzanoWeierstrass theorem. But Sikorski [9] contrived to prove a generalized theorem of Bolzano-Weierstrass for Q,: Every bounded 0)o-sequence of 0, contains a convergent (o,-subsequence. From this theorem follows in particular (cf. [8]), that every bounded monotonic 0o,-sequence of Q, converges, every o),-fundamental sequence converges, every Conest of intervals defines a unique element cx c=D 6, and every 0-cut (A, B) defines a unique element a E Q, (i.e. a :-5 ax _-:b for all a e A, b E B). These results mean that the transfinite rational domains Q, with the corresponding notion of an 0)6-sequence differ essentially from the finite rational domain Q with the notion of an 0o-sequence.

262

DIETER KLAUA

Defining for every a, b E Q, the number p(a, b) - Ia - biI Q, as the distance of the points a, b, the ordered pair (Q0, p) is a metric space of type co. By the accompanying concept of entourage one gets the open and closed sets X C Q, and for the system t of all open sets is (Q,,, t) a topological space of type (o, i.e. for every set 9Y C t it is valid: (1) (2)

(3)

9Y

A card

<

xoCý

v

t,

U 9Y r t,

0, 0,o E t.

The separation theorem means: For all non-void disjunct closed sets

X, Y C (Q there are disjunct open sets U D X, V D Y.In [6] we proved some theorems by an additional covering axiom. We get these theorems also from the generalized theorem of Bolzano-Weierstrass (see [8]), namely the following transfinite versions of known principles of the classical real analysis: INTERSECTION THEOREM OF CANTOR. Every descending %oosequence of non-void bounded closed subsets of Qo possesses a non-void intersection. COVERING THEOREM OF HEINE-BOREL-LEBESGUE. If U(x, ex) is an entouragefor every x E M of a bounded closed set M C 0, there is a subset X C M with M C UxF, U(x, s,) and card X < X, ACCUMULATION POINT PRINCIPLE OF BOLZANO-WEIERSTRASS. For every bounded set M C Qo with card M = X~, there is an accumulation point of M. The use of nonstandard topological concepts of Zakon [10] allowed Cantini [1, 2] to extend the algebraic and topological structure properties of Co, now Qo, from [6]. 3.

CUT NUMBERS

Now we turn to the real extension of Q, This is not superfluous; for in spite of the partially real character of Q( implied by the generalized theorem of Bolzano-Weierstrass, and in spite of the fact that there are more than countably many numbers 4 E Q, between any two rational

263

RATIONAL AND REAL ORDINAL NUMBERS

ordinal numbers a, b E Qa a < b, e.g. between 1 and 2, a number like

"ý2 does not exist in Q,. We define R, as the set of all cuts (A, B) in 0. such that the lower class A has no maximum (i.e. for every a E A there is an x E A with a < x). In Ro, a total order _-5,< is defined in the following way: If A = (A1, A 2 ) E R, and B =(B 1 , B 2) r R, then A :-5 B ý=# AC B1 ,

A < B <=* A1 C B,.

For (XE Q,, replace the cuts (]-->, ax [, [cX, -)->DE R simply by cx, so < of R, agrees on QU with the old that Q, Q Ro, and the order _-5, order 5, < of Q. Define l• = R,\Qo. For oc, P e R, with cx = f3 one defines the intervals: (ac, P3)= {( (i•cocR

:-

13},

[c, 3] = (a, 3) nQo,

)cx, 13(= {1

RIc < 4 < 13},

]x, 13[= )c, 3(n

,,

and similarly (oc, 13(, )oc,P3), (cx, -4(, )oc,-- (, )+-, 13), )<--, 13(, [oc, 13[, ... ]<--, 1[. Finally we define for cc, 13e R,: S.

a + 13P= ty(y E R A ]<--, y[ = {x + ylx, y E Q, A x < a A y < 1}), eyR, A ]+-, y[ = {-xlx E Qo A a < x}), -a = t E cx- 3 = (a + (-P) = t(y Ro A ]<-,-,[ = {x - ylx, y E , A x < cc A P < y}). On Qo, these operations agree with the old operations of Q. For (Xe R, and S(a) = ({x E Qolx < a), {x E Q01C :-5 x}),

it is the case that cc E Q0 if and only if S(a) is an 0-cut in Q,. Every cut in Ro, defines a unique element a E R,. For c, 13, 8, E R, we have: (a)

c+ 13=3+ (X, (c + 13)+ Y= c + (P +), a +0= a, -(-(X)=ca, oc- a50, a EQ, =>(X-ca=0, -(a +l3)=-OC-I3 (X e Q" V

(b)

(c)

13G Q"

-(c

+

1) =- c-13.

-(X El, , = -(X•Q Ee,•0 , XE :, C+ Q+ 4 E Q,, ac E e0. a e Q: a + E 0 <1J3E

C• cE

P

a < 10't- -P < -a, a = 1< * -oc

-•

264

DIETER KLAUA

(X ----P3<=-->-P----(X,

a < 0 0,

a > 0 <-cX < 0, a=O: a-(X=0, a :_5 0 ca-c- 0 a_= 0 a-a : 0. (d)

o: :_5

(e)

a <13 A y < 8 If y e Qo then

(f)

A 7 _5= > c+ Y :_5+, => + y-< p +

8,

a<13caa+y<13+y, a= 13 ca a+y= 13+y, a_ 13 a+y < +++y. aE Q,
= a for a 0 • e R,o a -+ x a + 1/i = a for a 1 _5_ E N,.

=

x for a


4.

INTERVAL COMPONENTS

In order to have good arithmetical properties, R, must be suitably restricted. The most obvious restriction is, instead of the domain R, of all cuts of Q0 (without maximum of the lower class), to consider only the subdomain C, of all 0-cuts of QO (without maximum of the lower class). For, for every a E G, the cut (]+--, o:[, [(x, -- [) has diameter 0, i.e. it is an 0-cut. Therefore one would expect that, as in the classical construction of R from Q, the 0-cuts of Q, with their diameter 0 should be the most suitable real extension of Q, But we know that every 0-cut of Q,, defines an element of Q, itself. Whence the construction of C,, (as in reference [4]) is superfluos. C,, is no genuine extension of Q,. A genuine real extension of Q,, is possible only by adjoining cuts of positive diameter (as in reference [51). We assume that the diameter of the cuts will agree with the diameter of the underlying 'transfinite rational number things', as with the 0-cuts and the numbers of QC,which

265

RATIONAL AND REAL ORDINAL NUMBERS

all have diameter 0. So we are led to the problem of finding appropriate partitions of Q,, whose elements are subsets of Q0 with a fixed positive diameter, which possess a real completion by cuts of the same diameter. The non-Archimedean structure of 0, indicates a possible solution. For, for every fixed 0 < h E QO, the system of sets A. = {Kh(x)Ix E Q(,, with Kh(x) = UnEN[X - nh, x + nh], is a natural partition of Qa, whose members have the same diameter which depends on h. Based on those partitions, we develop in the following a real extension of Q0, with good arithmetical properties. First of all we give some comments related to the partitions used. For every nonempty set M C R(, (in particular M Q (Uo,) bounded from above or below, the least upper bound sup M E R6 resp. the greatest lower bound inf M E R, exists. For nonempty sets X, Y C Q,, the number p(X, Y) = inf{Ix

-

yj Ix E X, Y E Y} E R

is called the distance of X, Y. For nonempty, bounded sets X C Q', the number 8(X) = sup{Ix - YJ Ix, y E X}I

R6

is called the diameter of X. For every oa r R, 8(a) = p(]<-, a•[, •a, -4[) = p(]<-,

x[, [a, -[)

is called the diameter of the number a. For 0 :_5 h e Q_ may be 8(h) = sup{nhln E N},

E(h) = inf{hlnl1 -_ n

E

N}1.

Finally, we call two numbers a, b E Q, of same magnitude if 8(lal) = 8(lbl),

or equivalently,

&(IaI) = _(IbI).

This relation is an equivalence relation in Qo, and for every 0 < h r Q, the degree g E Z/, of h (definable by Cantor normal forms, cf. [9]), and for h* = o• it holds that 6(lhl) = 8(h*). Then the numbers of the set H* =

i{0ig E Z,} C Q.

represent one-to-one all values 86(h) for arbitrary 0 < h E Q,. H* is order isomorphic with L, and every h E H* has the predecessor h/Wo E H* and the successor hWo E H*. Let us write H* = H* U {0}. Now we define for every 0 :_5 h E Q, and x e Q,:

266

DIETER KLAUA

Kh(x) = Une[NIX - nh, x + nh],

qh

= {Kh(x)IX E U,

R* = t-oShEQah = UheH'oh.

A, is a partition of QG, namely the system of all h-components of Q" for some notion of h-connectivity (cf. [6]) which may be defined in Q6 (generalizing the usual topological connectivity). The one-to-one family (Ah)h,,H is a partition of A*. For every h E H* and x E K E Ah, we have: K = Kh(x), 8(K) = 9(h), K= ]inf K, supK[, infK=x-8(h) sup K = x + 8(h) E 0.

I,

Hence the h-components K

E Ah (h E H*) are transfinite rational intervals in R0, and they are in Q, open-closed convex sets. We call them the (h-)interval components. They can be visualized as linear transfinite rational number balls of diameter 6(h). Now, as Q is used to define the classical real number domain R, so the system A'* forms the rational basis for the construction of real ordinal numbers with positive diameter. Of fundamental importance for this real extension of A'* is the following

THEOREM. For h

E

H*, h' = hwo, and a E K'

9h(K') = {K r

RhIK

C K'} = {K

E

E

A,,, the partition

thIK f K' • 0}

of K' into h-components satisfies: &ih(K') = K E jhla + rh' E K for some r E Q} = {Kh(a + rh')Ir e Q). Moreover,for each K e 1h, there is at most one r - Q with a + rh' E K, or, equivalently, for arbitrary r, s > E 0 we have: Kh(a + rh') = Kh(a + sh') €* r = s.

Proof. The proof is given in reference [7]. In the special case h = 1/co, h' = 1, and 0 E K' = K 1(0) = ]-4(1), 6(1)[ the theorem implies ViJ(]-8(1), 8(1)[) = Kll.(r)lr E

(Q.

Hence for every finite rational ordinal number x (i.e. x e Q(, with

267

RATIONAL AND REAL ORDINAL NUMBERS

-8(1) < x < 8(l), i.e. lxi < n for some n E N) there is a unique infinitesimally close classical rational number r, i.e. a unique r E ( with x E K1, 0(r), or, equivalently, Ix - rl < 8 (1/o) (i.e. Ix - rl < (n/co) for some n r N). 5.

REAL ORDINAL NUMBERS

The representation qh(K')

= {Kh(a +

rh')Ir r Q}

for h E H*, h' = hno, and a E K' E Ah suggests that we interpret the h-components K E A, as transfinite rational interval numbers of positive diameter 6(h), and that we extend Rh continuosly to the domain of real ordinal numbers of diameter 6(h) by forming cuts of the same diameter. Since Ro already fills all gaps of Qo, the real ordinal numbers can be immediately defined as appropriate elements of R,. Herewith every K E Ah is interpreted as inf K E R,. For simplicity, we also regard the elements of Q, as real ordinal numbers (of diameter 0). For K1, K 2 E A* we define the order <, K1 < K

K,

_5

2

K2

:

3x(x e K1 A Vy(y r K 2 := x < y)), K 1 < K 2 V K1 = K 2 .

Then <, _-5 is an irreflexive resp. reflexive total order relation in A'*. For every h r H0* is defined an h-cut of 0,, as an ordered pair (A1, A 2 ) of sets with: (1) (2) (3)

A1 U A 2 =QX < y for all x E A 1, y E for every x r_ A1 there is a y r A, with x + 6(h) < y;

A 1, A 2 •# 0,

A 2;

p(A 1, A2) = 9(h)

Then we define for h - H* and by using the above _-:in A*: R' = {a E R6 IS(a) = (]--, c[, [a, -[) is an h-cut), 0' = {[x E RoS(ca) is an h-cut with first K E Sh, K C [a, -- }, 0' = I C RolS(a) = is an h-cut without first K E %n,K C [a, -4[}. These sets are the sets of the h-real, h-rational, and h-irrationalordinal numbers of type co0.Finally we define:

R*= Ui.HsRh,

Qt = uhrsof

te

= .

is

ao

=.

R*' is the set of the real ordinal numbers of type (o, It is 0'~= Q

268

DIETER KLAUA

0' = 0, R' = Q,. We have Q, C Q C R* C Rl, and the domain A* is embedded into R* (namely onto *) by regarding K E A* as inf K. For every at E R, it holds that ca - a = -8(a) and for every a, P3 E R* with 6(c) = 8(p3):

S+ P3= a '3=

-8(a),

a + 3=-8(a) <=

-cc - 5(a).

These properties are the reason to define for all a, P3E R* and h E H* the zeroes, negative numbers, and differences: Oh

=-(h), *a = - - 8(c), = Oh for that h E H/4

0(a)

with

ot*=a aC

+(*1),

Rh.

If we define from now on - as ,, we get for every h E H* that, for example, every number structure (RG, +a - _) (Qh, :-,) forms a totally ordered commutative group. In general, the real ordinal numbers of R* have better algebraic and topological properties than the elements of R,. R is isomorphically embedded in R*. For those further results we call the readers attention to reference [8]. APPENDIX I

Various axiom systems of set theory have been developed for the settheoretical presentation of mathematics, including large sets; see L6vy [18], KMihnrich [16], and the literature cited in [16]. We give here the axiom system from [14] (or from the appendix of [13]), which is oriented by [11, 12] (and elaborated as a textbook [15]) and which is a modified ZFC set theory with the possibility of an a-prioridefinition of the rank relation. The axioms used are: I. Axiom of urelements, II. Axiom of extensionality, III. Axioms of separation, IV. Axiom of pairing, V. Axiom of union, VI. Axiom of power, VII. Axiom of restriction, VIII. Axiom of choice, IX. Axiom of universe. The Axioms I-IX serve for the presentation of the whole of mathematics (including category theory), while the elementary axiom I-VIII

269

RATIONAL AND REAL ORDINAL NUMBERS

suffice for the conventional mathematics (i.e. without hyperclasses and large classes and without any postulate of replacement). An urdomain U of urelements (e.g. a set of concrete, material things; U may be empty too) and the cumulative hierarchy of all sets (including large sets) above U form the real background of the axioms I-IX. Urelements and sets together are called objects. The objects, arranged in 02 stages over U, already form a model for the elementary axioms I-ViII. The basic concepts are variables for objects and the signs: E

U

-,

A

V

*

=•

V

3

3!

3!!

t

()

=

(3!, 9!!, t mean 'there is at most one', 'there is exactly one', the definite article 'that which'). We use Latin, Greek, and Fraktur characters as variables for objects. (For a theory without the admission of urelements, the nonlogical concept E suffices instead of e, U.) By the basic concepts, the expressions and terms of set theory are defined as usual. We define for objects a: a urelement <.* a E U, a set t-* a 0 U, a set system 4-* a set A Vx(x E a = x set). Axiom I. Axiom of Urelements

Va(a e U * -1 3x(x E a)). Axiom H. Axiom of Extensionality VAVB(A, B set A Vx(x

E

A . x E B) * A = B).

We define for sets A, B: A C B 4-* Vx(x E A • x E B), A C B t A C B A A

•

B,

Then, for sets A, B, C holds: A C B cA C B V A = B, A C B cA C B A 3x(x E B A x 0 A), ACA, ACBCC='ACC, ACBCA4:-A=B, AIA, ACBCCVACBCC=*ACC, ACB B TA, ACB B (t A.

270

DIETER KLAUA

Axioms III. Axioms of Separation (V)[M set = 3A(A set A Vx(x E A 4 x E M A P(x)))]. (P or P(x) is a set theoretical expression with free variable x. The variable A does not occur in P. (V) means generalization.) In case of existence

let: {xlP(x)} = tA(A set A Vx(x r A t* P(x))) (x is free in P, and A does not occur in P). For every set M the set {x E MIP(x)} = fxlx ( M A P(x)} exists (x is free in P). It is the urdomain U = {xlx urelement}, and there is the empty set 0 = {x E UIx # x} = tL(L set A -,3x(x E L)). It is 0 0 U, for objects a, A we have: a E U V a= 0 4=--,3x(x - A), A set - A =0 V 3x(x E A), and for sets A we have:

0 C A, A 0• -0•0 C A t-- 3x(x e A), A = 0 <* -3x(x E A). Axiom IV Axiom of Pairing For every a, b we have: 3A(A set A Vx(x E A (=- x = a V x = b)). Then, for all a, b, the following sets exist: {a, b} = {xlx = a Vx

=

b},

{a} = {a, a}

=

{xlx

and the ordered pair (a, b) = {{a, b}, {a} 1. Axiom V Axiom of Union For every set system M we have 3A(A set A Vx(x

E

A

4--

3X(X E M

Ax

E X))).

=

al,

RATIONAL AND REAL ORDINAL NUMBERS

271

Then, for every set system M, the set exists: UM = {xJ3X(X E M A x E X)}, and in case M # 0 also the set: AM = {x E UMIVX(X E M =ý x = {xIVX(X E M = x E X)}.

E

X)}

For all sets A, B, the sets exist:

A U B = U{A, B) = {xlx e A V xE B), A n B= {xE Alxr B} = {xlxr A Axe B}, A\A = {x E AIx e B) = {xlx E A A x e B1. Axiom VI. Axiom of Power For every set A we have:

3P(P set A VX(X

E

P 4* X set A X C A)).

Then, for all sets A, B, the sets exist: P(A) = {XIX set A X C A}, A* = A U P(A)), A x B = {XIX E P(P(A U B)) A 3x3y(x E A A yE B A X = (x, y))) = {(x, y)IX A AyE B). We shall define the universal sets (all-sets, stage sets) by the universal chains (chains of universal sets), i.e. by the nonempty sections of the well-ordered collection of all universal sets. For this, a universal chain (all-chain) is defined as a set system K, for which, for arbitrary X, Y and sets M, holds: (1)

X, Y E K =* X C Y V Y C X

(2)

0#*M C K=

(3) (4) (5)

3X(X E

(comparability),

M A VY(Y E M = X C Y))

(minimum condition), U E K, Xe KA 3Y(X C Y E K) = X C X* E K A VY(X C Y E K) = X* C Y), 0#MC KA 3X(X e K A VY(Y E M = Y C X)) =ýUMe K.

Now for every object V let:

272

DIETER KLAUA

V universal set t-- 3K(K universal chain A V e K). V is a universal set if and only if V = UK for a universal chain K. We say that an object L is a limit domain, if

3x(x E L) A Vx3y(x E L

*

x

E

y r L).

Finally, a universal domain (all-domain) is a universal set, which is a limit domain. Axiom VII. Axiom of Restriction Va3V(V universal domain A a E V). By the usual concepts one postulates the Axiom VIII. Axiom of Choice VM3A(M set system A M disjoint A 0 e M = A choice set of M). Let a universe be an object V with: (1) (2) (3)

V universal set, 3L(L limit domain A L r V), Vf(f function A dom(f) E V A range(f) C V • range(f)

SV). The conditions (1), (2) are equivalent to (1) and (2')

N E V for the set N of natural numbers.

Condition (3) is the postulate of replacement for V. Axiom IX. Axiom of Universe 3V(V universe A VX(X universe = X C V)). Then there exists the largest universe V = t V(V universe A VX(X universe

=>

X C V)).

The sets X E V are the conventional or small sets of mathematics, the other sets X 0 'V are the large sets. The sets X C 'V are the classes,

RATIONAL AND REAL ORDINAL NUMBERS

273

the other sets X ! V are the hyperclasses. The hyperclasses are still arranged in 0)2 stages above V From the axioms I-VII, for every object a the existence of the universal set follows: us(a) = min [XIX universal set A a ( X], i.e. the C-first universal set X with a E X. This enables us to define a priori (i.e. without using ordinal numbers) the rank relation (stage relation) E , namely for arbitrary a, b: a E b ¢ us(a) C us(b), a El b €=us(a) = us(b) t- a E b A b E a, a b c=us(a) C us(b) <- a E b A a 1b b, '-

We have for objects a, b, c, A, B: (a) (b)

(c) (d) (e)

a E b E c a E c, a E bVb a, 3x(xeA) = 3x(x E A A Vy(y E A = x E y)), ae A a - A, 3x(x - B) A Vx(x E A = x - B)= A E B.

Hence it follows that all the usual properties of E are valid (cf. [11, 14, 15]). The rank relation yields in Appendix II the existence of ursets and super structures for Nonstandard Analysis. APPENDIX I1

One method in Nonstandard Analysis (cf. Zakon [20], Stroyan and Luxemburg [19], Laugwitz [17]) uses modified ultrapowers of super structures, a variant of the usual ultrapower construction of mathematical structures. In a fixed set theory (we take for example our set theory from Appendix I) the superstructures ( U, E) have been used up to now for sets U of absolute urelements, i.e. for sets U C U, where U is the urdomain. For building the super structure of an arbitrary set A, this set must be replaced by an equipollent set U C U. That means an additional postulate on the cardinality of U, namely the existence of a one-to-one mapping from A into U. To avoid this, we shall construct without any new postulate for every set A an equipollent urset U of relative urelements (not necessarily U C U, even U n U = 0), which

274

DIETER KLAUA

is the basis for a desired super structure (U , E) over U. These ursets have been considered at first in [13]. Let A be a set, (V,),<, the sequence of sets with for every n c N: V 0 = A,

V 0.l = V, U P(V.)

and finally W = U,,-
-,3x3u(x EU A x E u E U), 0•

U.

This means that the elements of an urset U are the relative urelements of the E -structure (V7, E) (where E is the usual element relation, restricted to 77 ). The structure (U , e), for an urset U, is called in Nonstandard Analysis the super structure over U. For every set A, the urset of A, namely. U(A) = {A U

}

A = {A U '0, a)Ia e A},

is equipollent to A, and it is proved in [13], with the help of the rank relation, that U(A) is an urset. (The proof in [13] holds also for ordered pairs (x, y) = {{x, y}, {x}} instead (x, y) = {{x, y}, x}.) We now construct the modified ultrapowers of super structures (V7, e) over arbitrary ursets U (not only over absolute ursets U C U like in the literature, for instance in [20]). The exact proofs of the following holdings have been given in [13]. Let U be any urset and a any ultrafilter on a set J (then J = Ua). The superstructure over U is (7', E), where 77 = U,,-V, with V, = U and V+ 1 -,,=V U P(V,). Let = U,,<(V,,) and for all f, g E (D: f "- g <-* {i E JIf(i) = g(i)} e g, f e=g <-* {i E J Ifti) E g(i)}I E, To every a e 77 belongs the constant function a = (a),,,. There is a mapping go with: (a) (b)

dom(go) = UJ, U C range(go), range(go) urset; for all f, g E U and a e U holds: go(f) = go(g) 4* f- g, go(a) = a.

For the sets U* = range(go) and To = {f E Fif d U), the function go : U -- U*, where UV C TPo, can be extended to the function 0g' : To -* U* by the definition for every f E TO:

RATIONAL AND REAL ORDINAL NUMBERS

275

0x•(f) = go(h) for one (every) h E VJ with f h. Then we have for the mapping J±•: (a') (b')

g' : TPo - U*, U C U* = range(jto), for all f, g E To and a e U holds: Ixo(f) = ROW(g€*f - g, gx•a) = a.

U* urset;

THEOREM. (a) There is a unique mapping g : (D every f E D: (f)=

{g(f), iffE WVo (h)Ih e A h ( f}, iffi

{I

-

with for

Vo.

(b) For all f, g E (D holds: gi(f) = p.(g) 1* f-

g,

gx(f) e p.(g) 1:* f e g.

We call the mapping g the internal set function with reference to U, a, go. For every f e 4D resp. f E 4)\To, the value Ri(f) is the internal object resp. the internal set induced by f. The set *M= {Ig(f) IfE 0) C '-07'

resp.

*M\7--

is the set of internal objects resp. internal sets with reference to U, •, go. 7--*\*M is the set of external sets. The e-structure (*M, E) is the modified ultrapower of (U', E) with reference to a, go. Schinnrainstraj3e 15, D-76227 Karlsruhe, Germany REFERENCES 1. Cantini, A.: 1974, Ph. D. Thesis, Florence.

2. Cantini, A.: 1979, 'On non-Archimedean Structures of D. Klaua', Math. Nachr., 87, 35-41.

3. Cohen, L. and Goffman, C.: 1949, 'A theory of transfinite convergence', Trans. Amer. Math. Soc., 66, 65-74. 4. Klaua, D.: 1959/60, 'Transfinite reelle Zahlenrdiume', Wiss. Zeitschr.der HumboldtUniversitat, Berlin, Math. -Nat. R., IV, 169-172. 5. Klaua, D.: 1960, 'Zur Struktur der reellen Ordinalzahlen', Zeitschr.f. math. Logik und Grundlagen der Math., 6, 279-302.

276

DIETER KLAUA

6.

Klaua, D.: 1961, 'Konstruktion ganzer, rationalerund reeller Ordinalzahlen und die diskontinuierliche Struktur der transfiniten reellen Zahlenrciume, Berlin. 7. Klaua, D.: 1980, 'Interval components of non-Archimedean number systems', in Karl L. E. Nickel (Ed.), Interval Mathematics (pp. 129-143), Academic Press Inc. 8. Klaua, D.: 1981, 'Inhomogene Operationen reeller Ordinalzahlen', Archiv f. math. Logik und Grundlagenforschung,21, 149-167. 9. Sikorski, R.: 1948, 1950, 'On an ordered algebraic field', Soc. Sci. Litt. Varsovic, Sci. Math. Phys., 41, 69-96. 10. Zakon, E.: 1969, 'Remarks on the nonstandard real axis', in: Applications of Model Theory to Algebra, Analysis and Probability(pp. 195-227), New York.

Appendix 11. 12.

Klaua, D.: 1979, Mengenlehre, Berlin-New York. Klaua, D. 1981, 'Eine axiomatische Mengenlehre mit grtifltem Universum und Hyperklassen', Monatshefte fUr Mathematik, 92, 179-195. 13. Klaua, D.: 1986, 'Zur Existenz der Superstrukturen in der Nichtstandard-Analysis', photostat manuscript. 14. Klaua, D.: 1987, 'A-priori Konstruktion der Stufenbeziehung in einer modifizierten ZFC-Mengentheorie', photostat manuscript. 15. Klaua, D.: 1987, Exakte Grundlegung der Mathematik, Karlsruhe. 16. Kiuhnrich, M.: 1978, 'Superclasses in a finite extension of Zermelo set theory', Zeitschr.f math. Logik und Grundlagen der Math., 24, 539-552. 17. Laugwitz, D.: 1978, Infinitesimalkalkiil, Mannheim-Wien-Ziirich. 18. Ldvy, A.: 1976, 'The role of classes in set theory', in G. H. MUller (Ed.), Sets and Classes (pp. 173-215), Amsterdam-New York-Oxford. 19. Stroyan, K. D. and Luxemburg, W. A. J.: 1976, Introduction to the theory of Infinitesimals, New York-San Francisco-London. 20. Zakon, E.: 1974, A New Variant of Non-Standard Analysis, Lecture Notes in Mathematics, Vol. 369, pp. 313-339.

INDEX OF NAMES

Dawson, J., xv Dedekind, R., viii-xi, xiii-xvi, xviii-xxi, xxiii, xxvi, xxviii, 13, 16-17, 19, 93, 97, 101, 102, 108, 133, 141-142, 170-171, 184(18), 194, 197-203, 208, 225, 239 Dehn, xxv, 172 Desargues, 158-159, 161, 175 Descartes, R., vii, 17 Dieudonnd, J., 203 Dino, S., 126 Du Bois-Reymond, P., ix-x, xv-xvi, 9, 13, 107, 132-133, 136, 169-170, 177-178 Dummett, M., 31

Abel, N. H., 6-7 Ackermann, 246 Archimedes, 8, 133, 149, 152, 155-157, 163-164, 166, 170, 174-175, 198-200, 207 Aristotle, 17 Arkeryd, L., 234 Artin, E., xxvi, 191-193, 195-197, 199-200, 203 Bachmann, F., 111 Banazchewski, xxi Berkeley, G., 10 Bernays, P., 111 Bernouilli, G., 169 Bettazzi, R., xvi, xx-xxi, 107-108, 110, 136 Bindoni, 172 Bishop, E., xi-xiii, 30, 32, 34, 37, 52 Boltzmann, 234 Bolyai, 147, 172, 176 Bolzano, B., 184(11), 194, 196, 207, 261-262 Borel, E., 7, 107, 136, 223 Bourbaki, N., xxiv, 136, 199 Bridges, D., xiii, 37, 52, 65, 68, 70, 90 Brouwer, L. E. J., xi-xii, 31, 34, 90(5)

Ehrlich, P., xxviii Enriques, F., xxv, 184(16) Euclid, vii, 8, 119, 132, 147-149, 152, 154-155, 161- 162, 164, 166, 172, 175-176, 181 Eudoxus, vii Fermat, 32 Fiedler, W., 126 Fisher, xxiii Fontenelle, 169 Gauss, K. F. 134, 169, 178 G6del, K., 31, 131, 136, 208 Goffman, C., xxi, 264 Goldbach, 32 Grassmann, 178, 180

Cantor, G., viii-xi, xiii, xv, xviii, xxiii, 13, 16-21,24,59, 100, 108, 110, 125, 127, 133, 135-136, 141-142, 148, 155, 169-170, 177, 191-192, 194, 197, 207, 227, 259, 265 Cauchy, xiv, 29, 42, 50, 52, 57, 102, 180, 194-195, 198, 200, 203, 261 Cavallieri, 8 Cayley, 175-176, 180 Chevalley, C., 203 Cohen, L., xxi, 264 Conway, J. H., xiii-xv, xxvii-xxviii, 210, 239-241,246-248, 251,255-256, 257(1,2) Cutland, N. J., 234

Hahn, H., xvi, xx, xxii-xxiii, xxvi, 107-113, 125, 127, 172 Hamilton, W. 148, 161 Hankel, H., ix Hardy, G. H., xv, 136 Hasse, H., 194, 201 Hausdorff, F., 107, 225 Hauschild, K., 200 Heine, H., 13 Helmholtz, H. von, 134-135, 152, 165, 167,

277

278

INDEX OF NAMES

174-175, 180 Hessenberg, xxviii, 259 Hewitt, E., xxvii, 130, 208 Hilbert, D., xvii, xxii-xxvi, 108, 111, 127-131,136, 148-150, 152-155, 157-159, 161, 163-167, 172, 175, 191, 193, 197, 199, 201, 203(3) Hobson, E. W., ix, xi H61der, 0., xvi, xix, 185(18) Jevons, S., 150 Jordan, C., 202 Kant, I., 128, 130-132, 175, 181 Keisler, H. J., xxi, xxvii Killing, 127 Klaua, D., xxviii, 259 Kleene, S. C., 31 Klein, F., 126, 129, 132, 178 Kronecker, 13 Kuratowski, 99 Lagrange, L.-J., xxvi, 10, 196-197, 200 Laugwitz, D., xxi, 107, 125, 196, 208 Leibniz, G. W., 8, 10, 17, 148, 169, 207 Levi-Civita, T., xix, xxiii, xxv, 107-108, 113, 125, 127, 172, 185 Lie, S., 134-135, 166-167, 178 Lobachevsky, N. J., xxiv, 130, 147-148, 154, 163-165, 167, 172, 176 Locke, J., 17 Loeb, P. A., 221-223,227 Loria, G., 134 Luxemburg, W. A. J., xxii Malcev, 208 Mandelkern, M., 70, 76 Marion, xxiii Mill, J. S., 175 Minkowski, 202

Pascal, B., 8-9, 126, 134, 162-163 Pasch, M., xvii, 132, 140, 180, 199 Peano, 93, 97, 127, 150 Peirce, C. S., x Pfister, A., 202 PoincarO, H., xxiv, xxv, 7, 109, 169, 176, 186 Poisson, 169 Priess-Crampe, S., xxi Reeb, 234 Richman, 65, 68, 70 Riemann, B., xxiv, 4, 11, 32, 134-135, 147-148, 154, 163-165, 167, 176, 180 Roberval, 8 Robinson, A., xv, xviii, xxi-xxii, xxvi-xxvii, 133, 208-209, 216, 223, 230, 232 Rogers, H., 67 Rolle, M., 196 Russell, B., xiii, 181, 186 Schepp, A., 107, 130 Schmerl, J., xxi Schmieden, 208 Schoenflies, A., xix, xxii, 107, 125, 127, 172 Schreier, xxvi, 191-193, 195-197, 199-200, 203 Schur, F., xxv, 158, 172 Scott, D., xxi, 200-201 Sikorski, R., xxviii, 259, 261 Skolem, T., 208 Specker, E., xiii, 50, 67-69 Spinoza, B., 17 Steinitz, 191, 197 Stevin, S., vii Stieltjes, 7 Stolz, 0., xv-xvi, xvii, 107, 136, 170-171, 184(15) Stromberg, K., 130 Sylvester, J. J., 202

Ockam, W. 98-99

Tannery, 18 Tarski, A., xxvi, 212-213, 228 Taylor, I1 Thomae, J., xv Torretti, R., 133 Troelstra, A. S., 31

Padoa, 150

Unger, L., 128

Nelson, E., 231-232 Neumann, B. H. von, xxi, xxviii, 93, 239, 241,254 Newton, I., vii, xvii, 8, 10, 175, 178

INDEX OF NAMES Vahlen, 172 Van Dalen, D., 31 Van den Berg, I. P., 234 Veronese, G., x, xvi-xviii, xix-xxiii, xxv, 16, 107-108, 113, 116-119, 121-123, 125-130, 133, 135-138, 141, 183(2), 184(17), 185(18) Vesley, 31 Vivanti, 127, 136 Wallis, J., vii, xvii

279

Weierstrass, ix, xiii, 11, 13, 170, 178, 203, 208, 261-262 Weil, A., 194, 195, 200 Wundt, W., 132 Zakon, E., 230 Zeno, 233 Zermelo, 99, 109, 227-229, 231 ZOllner, 180, 187