Backgrounds of Arithmetic and Geometry: An Introduction

Backgrounds of Arithmetic and Geometry An Introduction

SERIES IN PURE MATHEMATICS Editor: C C Hsiung Associate Editors: S S Chern, S Kobayashi, I Satake, Y-T Siu, W-T Wu and M Yamaguti

Part I. Monographs and Textbooks Volume 10: Compact Riemann Surfaces and Algebraic Curves Kichoon Yang Volume 13: Introduction to Compact Lie Groups Howard D Fegan Volume 16: Boundary Value Problems for Analytic Functions Jian-Ke Lu Volume 19: Topics in Integral Geometry De-Lin Ren Volume 20: Almost Complex and Complex Structures C C Hsiung Volume 21:

Structuralism and Structures Charles E Rickart

Part II. Lecture Notes Volume 11: Topics in Mathematical Analysis Th M Rassias (editor) Volume 12: A Concise Introduction to the Theory of Integration Daniel W Stroock Part III. Collected Works Selecta of D. C. Spencer Selected Papers of Errett Bishop Collected Papers of Marston Morse Volume 14: Selected Papers of Wilhelm P. A. Klingenberg Volume 15: Collected Papers of Y. Matsushima Volume 17: Selected Papers of J. L. Koszul Volume 18: Selected Papers of M. Toda M. Wadati (editor)

Series in Pure Mathematics - Volume 23

BACKGROUNDS OF ARITHMETIC AND GEOMETRY An Introduction

Radu Miron Dan Branzei University Al. I. Cuza Romania

World Scientific

Singapore • New Jersey • London • Hong Kong

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BACKGROUNDS OF ARITHMETIC AND GEOMETRY — AN INTRODUCTION Copyright © 1995 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA.

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PREFACE There is a general opinion that the foundations of Mathematics make up a part of Mathematics completely from the remaining ones, which only a small number of researchers will be interested in. In supporting this opinion, some arguments have also been put up: natural numbers have been efficiently used for millieniums before being defined on the basis of Peano's Axiomatic system. Real numbers had been achieved but in the Middle Ages, sets had taken up the role of foundation of Mathematics on the basis of Cantor's naive set theory long before the edification of a rigorous logical theory. This opinion is profoundly mistaken. First of all, in present-day Mathematics it is not possible to make a delimitation, not even a vague one, of its foundations Secondly, the efforts meant to found Mathematics have appeared very early, Euclid's monumental work constituting an undeniable example We can say that Mathematics evolved in close relations with its foundations, and to know this fact is an essential element of general culture. The Romanian Mathematician Dan Barbilian, also known as an outstanding poet under the name of Ion Barbu, sy nthetized this truth in a remarkable assertion for Mathematics, History and Philosophy of Culture: Homer is not the obligatory gate through which one can approach the Greek world. Greek Geometry is a wider gate, from which the eye can look upon an austere, but essential landscape. School's aim is not that of acquiring knowledge, but that of achieving creative thinking capacities, a non-dogmatic and non-prejudiced one. If Mathematics could be torn from its foundations, it would became a series of formulae, receipts and tautologies that could not be applied any longer to the objective reality, but only to some rigid, mortified schemes of this reality. The teaching of Mathematics expresses a remarkable balance between Mathematical concrete constructions and their rigorous logical founding. But balance is dynamic, variable in time and space, the teacher presents not only the contents of a theory, but its relevance in Mathematics and life, as well. Likewise, theoretical Informatics can not be thought separately from the foundations of Mathematics. The foundations of Mathematics constitute an ample domain too for only a book. In order to achieve a relevant prospect we chose Arithmetic for the constructive method and Geometry for the axiomatic one. The alternation of the two methods ensures a unitary understanding of elementary Geometry and of its organic link to the study of real numbers. Here is a description of the contents of the book The first chapter presents the naive set theory with the constructions and structurings of the families of cardinal and ordinal numbers

v

vi The second chapter presents the natural numbers by means of the Frege-Russell constructions, then successively extending N to Z, Q, R - through the factorization of the ring of Cauchy sequences by means of the ideal of null sequences - and C. The third chapter presents the axiomatic Theories differentiated according to their degree of formalization and the central metatheoretical problems. The subject is exemplified by the reconstruction of N on the basis of Peano's semiformalized axiomatics. The fourth chapter presents the algebraic backgrounds of Geometry. The problems of this chapter constitute a recent and important contribution of Romanian Mathematics, especially of the first of the authors of this book. A minimal Weyl type axiomatic system introduces the linear, affine and Euclidean spaces. The fifth chapter edifies Geometry on the basis of Hilbert's axiomatic system Here, the becoming attention is paid to the deduction of some first consequences of axioms; the way from these to the main Euclidean results is assumed to be know. The metatheoretical analysis points out the arithmetical model i.e. the leading ideas of the analytical Geometry of the space. In the sixth chapter, Hilbert's axiomatics is correlated to that of Birkhoff. quite recently adopted to the didactics of Geometry in many countries. The seventh chapter presents in brief the mean types of geometrical transformations, their actions upon the fundamental geometric figures and the results of certain geometrical transformations compounding. The eighth chapter exposes in a modern manner Felix Klein's conception on Geometries and exemplifies it through the study of affine, projective and plane hyperbolic Geometries. Francisc Rado, a specialist of international rate in Geometries algebraic foundations, Barbilian spaces, finite Geometries, etc, has attentively studied the manuscript and made pertinent observations of great usefulness. Appreciating the quality of the material, he outlined a new unitary prospect to it by the ninth chapter in which he presents the construction of Geometry through an axiomatics of Bachmann type, with Isometry as a primary notion. The first eight chapters contain exercises or problems. The book benefits from a beautiful tradition of the city Iasi (Jassy) materialized by Professor Izu Vaisman's The Foundations of Mathematics (in Romanian, EDP, Bucharest, 1968), as well. The didactic experience of the authors, who have been teaching this discipline for over two decades at the oldest University in Romania has also benefited from an ample dialogue with specialists in this field who studied their work The Foundations of Mathematics and Geometry (in Romanian, Ed. Academiei, Bucharest, 1983). This work turns to account this experience, possessing a personal hint by its structure and conception The original feature is at its top in chapters IV, VIII and IX, but it is to be found in each paragraph, as well

vii The unity of the work, as a mirroring of the unitary character of Mathematics in general and especially of Geometry is evident. The book is quite accessible to high school pupils in the last forms, but its being taken for a handbook is out of question; first of all, its reading is supposed to follow a first systematic study of Geometry on the basis of an axiomatic system. Secondly, details of a technical character meant to point out their ideas and connections have often been omitted. The book is useful to the students in Mathematics faculties (for getting the master or in drawing up a doctorship thesis included), to the Mathematics teacher, to the researchers in Mathematics (even for those in very specialized domains), and to-other researchers that use the mathematical apparatus, to computer sciences, for instance. The authors consider that the reader will be convinced not only by the essential aspects ensuring the unity and vigour of Mathematics, but of the unaltered value of the geometrical measure with its remarkable elegance, as vsell. Acknowledgments. We express appreciation to Mr Mihai Eugeniu Avadanei and Miss Oana Brdnzei for their essential contribution in the improvements of English version We would like to extend special thanks to our colleagues Mihai Anastasiei and Adrian Albu, who brought along valuable comments and suggestions in the evolution of this text We are also grateful to Mihai Postolache, Valentin Clocotici, Radu Negrescu, Daniel Rameder, Bogdan Balica and Cornelia Ivasc who realised the computer-typed version of the manuscript Last but not least we are thankful to all our students who have been carefully studying our courses and conferences Their opinions decisively contributed to a better crystallized conception on the subjects didactics.

THE AUTHORS

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CONTENTS PREFACE

v

Chapter I. ELEMENTS OF SET THEORY §1. Set Algebra §2. Binary Relations and Functions §3. Cardinal Numbers §4. Ordinal Numbers Exercises

1 1 4 6 12 17

Chapter II. ARITHMETIC §1. The Set N of Natural Numbers §2. The Set Z of Integers §3. Divisibility in the Ring of Integers §4. The Set of Rational Numbers §5. The Set R of Real Numbers §6. The Set C of Complex Numbers Exercises

18 18 22 26 32 37 44 46

Chapter Til. AXIOMATIC THEORIES §1. Deductive Systems §2. The Metatheory of an Axiomatic Theory §3. Peano's Axiomatics of Arithmetic §4. The Relation of Order on the Set N of Natural Numbers §5. The Metatheoretical Analysis of the Axiomatic System of Natural Numbers Exercises

48 48 50 53 58 59 61

Chapter IV. ALGEBRAIC BASES OF GEOMETRY §1. Almost Linear Spaces §2. Real Linear Spaces §3. Real Almost Affine Spaces §4. Real Affine Spaces §5. Euclidean Spaces Problem

63 63 69 74 79 82 86

IX

x

Contents

Chapter V. THE BASES OF EUCLIDEAN GEOMETRY §1. Group I of Axioms §2. Group II of Axioms §3. The Orientation of the Straight Line §4. Group III of Axioms §5. Group IV of Axioms §6. Group V of Axioms §7. The Metatheory of Hilbert's Axiomatics Exercises

87 88 90 92 98 105 Ill 114 120

Chapter VI. BIRKHOFF'S AXIOMATIC SYSTEM §1. The General Framework §2. Axioms and Their Principal Consequences §3. Elements of Metatheory Problem

122 122 123 131 135

Chapter VII. GEOMETRICAL TRANSFORMATIONS §1. Generalities §2. Isometries §3. Symmetries §4. Vectors §5. Translations §6. Rotations §7. Homotheties §8. Inversions Problems

139 139 140 142 144 149 151 156 161 165

Chapter Vm. THE ERLANGEN PROGRAM §1. Klein Spaces §2. Plane Affine Geometry §3. The Real Projective Plane §4. Plane Projective Geometry §5. The Hyperbolic Plane §6. Complements of Absolute Geometry

167 167 169 174 183 190 204

Contents §7. Plane Hyperbolic Geometry §8. Hyperbolic Trigonometry Problem

xi 215 227 235

Chapter IX. BACHMANN'S AXIOMATIC SYSTEM (Francise Rad6) §1. The Isometries of the Absolute Plane §2. The Embedding of the Absolute Plane into the Group of Its Isometries §3. The Group Plane §4. The Consequences of Bachmann's Axioms §5. The Theorem of Perpendiculars and its Applications

236 236

HINTS

253

BIBLIOGRAPHY

261

INDEX

283

240 241 244 248

CHAPTER I ELEMENTS OF SET THEORY The set theory has a fundamental importance in present-day Mathematics. From the logicians' point of view, Mathematics is the theory of sets and of their consequences. In the following lines, the construction of natural numbers and of the other known numbers is based on the concept of set, and the notion of mathematical structure is considered as a set of elements subjected to a system of axioms. It is not easy to achieve a rigorous presentation of the notion of set. Its axiomatic description is made through a complicated process. That is why we place ourselves within the naive point of view here, according to which the notion of set is supposed to be known. §1. The Set Algebra We shall denote the sets by A, B, C, . . . . the elements o f a s e t b y a, b, c, ..., x, y, z, ... etc., and will use the logical symbols =, =», V, 3, etc., whose meanings we consider to be known. A set will often be characterized by certain properties of its elements. One writes {x: P(x)} , where P(x) is a proposition referring to x, this meaning the set of all those .r-es, so that P(x) is true. From this point of view, the terms "property" and "set" are synonymous ones. If x is an element of the set A , we will write xGA, while x £ A means that " x is not an element of the set A ". We shall admit the existence of a set, denoted by 0 , which has no elements. For any x, {x} denote the set containing only the element x. Analogously, {xt, x2, ..., xn } will be the set containing only the elements x,, x2, ..., xn (supposed to be distinct). Definition 1.1. Let be A and B two sets. We shall say that A is a subset ofB or that A is included into B or A is a part of B, if xGA =» xGB, and we shall write ACB. If ACB and BCA, we shall say that A and B are equal, and write A = B; to deny that we write A^-B. If AC B, A^ B, then A will be called a proper subset of B. This property is denoted by AcB. It follows that ACB means that Az-B, or A = B. It is easy to deduce from the definition of inclusion that:

2

Backgrounds of Arithmetic and Geometry. An Introduction

l.ACA (reflexivity); 2. ACB, BCC => ACC (transitivity); 3. ACB, BCA => A = B (antisymmetry). The antisymmetry property expresses the fact that a set A is completely characterized through the totality of elements belonging to it; for example: \"*l \ - * l » -*2 '

3*

4/

—

\

2'

4*

3*

I / '

Of course, for any set A it occurs that 0 C A , and it immediately follows that the void set is unique. Definition 1.2. The set A\JB = {x:x G A or x € B) is called the union of the sets A and B. This definition extends to a certain family of sets {/4,} ie/ and is written |J Ar is;

We draw the attention to the fact that we are using the term of "family" synonymous to that of sets in order to avoid annoying repetitions. Definition 1.3. The set Af\B of the sets A and B.

{A,},€l,

= {x:XGA

By extending that definition f]At = { j r : V / € / , * , £ / ) , } holds.

and xGB)

to

a

is called the intersection

certain

family

of

sets

One denotes by 7(A) the set of all A's subsets; one prefers for IP(/4) the name power set of A better than the set of parts of A. Obviously: AET(A),

0GT(A),

MCA

=> MGJ>(A).

Theorem 1.1. For any sets A, B, C the following properties hold: A\JB = B\JA Af\B = Bf)A (commutativity); A\JA = A Af]A = A (idempotence); A[}0 = A Af]0 = 0; A[)(B\JC) = (A\JB)\JC Af}(Bf}C) = (Af)B)f)C (associativity); ACA\JB Af)BCA; ACB=>A\JB=B ACB=>A[\B A. The proof of this theorem is immediate; we leave it to the reader. It is also easy

Elements of Set Theory

3

to show the following: Theorem 1.2 (The laws of distributivity). Af](B[jC) A[j(BnC) Definition 1.4. The set A\B

= =

(Af\B)[j(AnC); (A[jB)r\(A[jC).

= {x:xGA,

sets A and B. The set CAB = A\B, for BCA

x$B}

is called the difference of the

is called the complement of B with

respect to A. If A is fixed, it is called universal set. CAB is denoted by B and it is simply called the complement of B. One immediately deduces ACB*»BCA,A

= A,CA0

= A,CAA = 0, and

Theorem 1.3 (Rules of Pierce and De Morgan). A\JB Af]B;Af]B A{}B = =Af]B;AT\B

= = A\JB;

IM = fM;TH = IMi€/

/€/

ie;

1i e6 ;

The proof does not entail difficulties. From the last three theorems it follows the principle of duality in the set theory. From any general relationship between sets that makes the operation U, (1» C , D intervene, a new true relationship is obtained, replacing respectively the previous operations by f], [}, 3 , C and the void set 0 by the universal set A and vice versa. If the sets A and B have the property Af]B = 0 we say that A and B are disjoint sets or non-overlapping. Let be a and b distinct elements; the set {a, b) enables us to know this pair of elements without specifying which is "the first" and which "the second". We can favour one of these elements, a, for example, in order to consider it as being the first one, giving the set {{a, b), a) which we denote more briefly by (a, b) and call it

4

Backgrounds of Arithmetic and Geometry. An Introduction

ordered pair or couple. The symbol (a, b) is considered in the case a = b, too, but now it notes the set {a, {a}}. The couple {a, (p, c)) permits the identification of the elements a, b, c, but an ordering of them, as well; it is more simply denoted by (a, b, c) and it is called ordered triple. For a certain set A, the set A = {(x,A):x e A) is called associated with A. Let us notice that A = B => .4 = B and/I *B =>AC\B = 0 . Definition 1.5. Let {A^.

be a certain family of sets The set V Ai = \JAi is

called disjoint union of the sets of the family under consideration. If the family contains only two (distinct) sets A, B, for their disjoint union the notation A v His also used This notation is also used when A=B, but its meaning being now specified by A \/ B= A\JA. §2. Binary Relations and Functions The notions of relation and that of function are of a general interest. We will briefly expose them. If X and Y are two sets, then X x Y = {(x, y): xeX, yeY} is called their Cartesian product. Definition 2.1. A binary relation p is an ordered triple p = (X, Y, G), where X and Y are sets (called base sets ofp), and G czX x Y. (The set G is called the graph of the relation p.) Instead of the notation (x,y)eG, the notation xpy (which is read "x is in the relation p withy) is preferred. Of course, when the base sets, X, Y are implied in the context, the binary relation p is specified by its graph, G, but that does not mean that a binary relation would coincide with its graph. For the binary relation p = ( X, Y, G ) , the dom p = {x: 3y xpy) and codom p = {y. 3X xpy) are considered, sets which are called the domain and codomain, respectively, of the binary relation p. The binary relation p"' = (Y, X, G') characterized by yp ~lx xpy is called the reverse or the inverse of the relation p. Obviously, dom p"' = codom p and codom p'l = dom p.

Elements of Set Theory

5

Definition 2.2. Let p = (X, Y, G) and a = (Y, Z, H) be two binary relations; we call composition of these binary relations the binary relation a o p - (X, Z, K), where xa o pz ** 3y, By, xpy and yaz. An important particular case of binary relations consists of the homogeneous relations, when the two base sets coincide. Although for the binary relation p = (X, X, G), GCX2 occurs it is said that p is a binary relation on X. Among the homogeneous binary relations the relations of equivalence and order distinguish themselves as most important ones. Relations of Equivalence Definition 2.3. The binary relation p = (X, X, G) is an equivalence (on the set X), if it satisfies the following conditions: 1. it is reflexive, that is x £ X «■» xpx; 2. it is symmetric, that is xpy =* ypx; 3. it is transitive, that is xpy and ypz =» xpz. For such a binary relation, to an arbitrary element x G X its class of equivalence Px = {y- yPx) ' s associated. Frequently px is named coset of x modulo p. For the distinct elements x, y, the classes of equivalence px, p ate disjoint or coincide. Due to the reflexivity property, any element x from X belongs at least to a class of equivalence (xGpx),

hence (J px = X. xex

The set XI p = { px: x £ X } is called factor set or quotient set of X through the relation p; its elements are the classes of equivalence and make up a partitioning of X. The partitioning XIp is obviously uniquely determining the equivalence p. The map x: X -» XIp defined through ir(x) = px is called the canonical projection of the equivalence p or the quotient map of p. Relation of (Partial) Ordering Definition 2.4. The binary relation p',

is "any", fixed, and p&O,

and there

q.e.d.

Theorem 1.9 (The algorithm of dividing with remainder in N). 7f n, m are natural numbers with m ^ O .there exist unique natural numbers q (called quotient) andr (called remainder) so that n = mq*r and r< m. Proof. Let be A = {xlx £ N and mx < n); by the preceding theorem, A is a proper subset of N. We have 0 £ A and in order not to reach by induction the false conclusion A = N, there must exist q£A so that q' £A. From ^ £ / 4 there results the existence of an r £ N so that mq + r - n. Since q' £A, it results /n(^ + l ) > mq+r, therefore r <m. To prove the uniqueness of (q, r) let us suppose that there also exists (h, k)^(q, r) so that n = mh+k and k<m. There follows that h£A, h' $A. If, for example, h < q it follows / i ' < q and, therefore, w/i' < mq n in contradiction with /i' £ A. Analogously from q < h we would contradict q' &.A. Therefore, h = q. There follows n = mq+r = mq+k, therefore r = k, q.e.d. Remarks. The notion of natural number is obviously objective and non-contradictory. But here it is not a question of abstracting this notion from the objective reality but of constructing it within Mathematics. The fact that we can not prove the objective character of the natural number within Mathematics constitutes a secondary, non-edifying aspect. Frege's conviction is firm in this context: "Within Mathematics, we do not deal with objects we know as something extraneous, exterior, by means of our senses, but with objects given directly to the reason, which can fully intuit them as something characteristic, of its own. In spite of the above mentioned fact, or, better said, just due to it, these objects are not subjective airy visions. There is nothing more objective than the laws of Arithmetic" (see [172]). §2. The Set Z of Integers Starting from the set N of natural numbers, we shall construct the set Z of the integer numbers with the idea of completing the additive semi-group ( N , +) so that we may get a group on the new set of numbers. Let us consider "K- ( N x N ) . We define a binary relation ~ on "Kby: (m, n) ~ (mx, n{) *» m+/», = m , + n .

Arithmetic

23

Proposition 2.1. The relation ~ is an equivalence on the set !K Indeed, (m, n) ~ (m, n) occurs because m + n = n+m. From(m, n) ~ (m,, n,) it results (/w,,«,) ~ ( m , « ) , because w»+n, = w, +n =» /n, +n = m+nr We also have [(m,n)~(ml,ni) and (m,, n,) - (m2, n2)] =» (m, n) - (m2, n2), q.e.d. Proposition 2.2. The following properties hold:

1. Vm G R, ( m , r o ) ~ ( 0 , 0 ) ; 2. (m,n) ~(m-n,0) for m> n; 3. (wi,fl)~ ( 0 , n-m) for n> m. The proofs of these properties are immediate. Let be Z = W —;; an element of this set is called integer number, or simply integer, and Z is called the set of the integer numbers. As Z is the set of the classes of equivalence related to — , we will denote by [m,n] the integer number determined by (m,n). Proposition 2.3. If (m,n) 1. (m+p,n+q)~(m]+p],nt

2.

— (mt, «,) and (p, q) ~(p,,q,), + a,);

then

(mp+nq,mq+np)~(mlpt+n]ql,mlqt+nlpt).

Proof. 1. From wi+n, = m^+n and p+tf, - pt*q m*p+n]+ql = WJ, +/>, +n*q. One analogously checks 2.

are immediately results

Proposition 2.4. 77ie applications + : Z 2 -* Z and ' :Z2 -* Z defined by [m,n]+[p, q] = [m+p, n+q] and [m, n] • [p, q] = [mp+nq,mq+np], respectively, do not depend on the representatives (m, n) and (/>, q) of the integers [m, n] and[p, q], respectively. Indeed, the preceding equalities are immediate consequences of Proposition 2.3. Therefore, + and • are binary operations on the set Z; we will call them addition and multiplication of the integers. Theorem 2 . 1 . The set Z of the integers together with the operations of addition and multiplication is a domain of integrity. Proof. 1) (Z, +) is an Abelian group. Indeed, [m,n]+[[p,q)+[r,s]]

= [ m , n]+[p + r, q+s] = [ m + a + r , n+q+s] = = [[m,n]+lp,q]]+[r,s].

24

Backgrounds of Arithmetic and Geometry. An Introduction

The neutral element is [ 0 , 0 ] , [m, n] +[0, 0] = [m, n]. The opposite-[m, n] of the number [m,n] is the integer [n,m] since the following equality holds [m,n]+[n,m] = [m+n,n+m] = [ 0 , 0 ] . Finally, [m,n]+[p,q] = [p,q] + [m,n]. 2) ( Z , • ) is a unitary, commutative semi-group. [ [ " » . « ] * [P. 9 l ] *[>.•*] =[mp+nq,mq+np] '[r,s] = = [mpr + nqr+mqs+nps,mps + nqs+mqr + npr] =[m,n] • [[p,q] ' [r,s]] [ 1 , 0 ] is the unit: [m,n] • [ 1 , 0 ] = [ « , » ] . Finally, [ / » , » ] • [ p , ? ] = [mp+nq, mq+np] = =[/>, 9] • [«» » ] • 3) Multiplication is distributive in relation with addition. [m, n] • [ [ p , ^ ] + [ r , s]] = [m, «] • [ p + r , g + j ] = [wip+wir+ng+n.f, m n; therefore there exists n*0 such that m = n+r and so [m, n] = [ r , 0 ] . The initial equality becomes [ r , 0] • [ p , q] = [ 0 , 0] that is rp = r ^ . There follows that p = q, that is [ p ,