i-TJ tj"" .i:*U H^"' 248 D29 1901 CORNELL UNIVERSITY LIBRARY BOUGHT WITH THE INCOME OF THE SAGE ENDOWMENT FUND GIVEN IN 1891 BY HENRY WILLIAMS SAGE MATHEMATICS Cornell University Library QA 248.D29 1901 Essays on the theory of "SSliiiiiili «"" '"""""^"a'oOI 586 282 ill =^A Cornell University Library The original of tliis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924001586282 IN THE SAME SERIES. ON CONTINUITY AND IRRATIONAL NUMBERS, and ON THE NATURE AND MEANING OF NUMBERS. By R. Dedekind. From the German by IV. W. Beman. Pages, 115. Cloth, 75 cents net (3s. fid. net). GEOMETRIC EXERCISES IN PAPER-FOLDING. By T. SuNDARA Row. Edited and revised by W. W. Beman and D, E. Smith. With many half-tone engravings from pho- tographs of actual exercises, and a package of papers for folding. Pages, circa 200. Cloth, Si.oo. net (4s. 6d. net). (In Preparation.) 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THE OPEN COURT PUBLISHING COMPANY 324 DEARBORN ST., CHICAGO. LONDON: Kegan Paul, Trench, Triibner & Co. ESSAYS THEORY OF NUMBERS I. CONTINUITY AND IRRATIONAL NUMBERS II. THE NATURE AND MEANING OF NUMBERS RICHARD DEDEKIND AUTHORIZED TRANSLATION BY WOOSTER WOODRUFF BEMAN PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF MICHIGAN CHICAGO THE OPEN COURT PUBLISHING COMPANY LONDON AGENTS Kegan Paul, Trench, TrUbnee & Co., Ltd. 1901 TRANSLATION COPYRIGHTED The Open Qz(urt Publishing Co. igoi. CONTENTS I. CONTINUITY AND IRRATIONAL NUMBERS. PAGE Preface . . . . i I. Properties of Rational Numbers . . 3 II. Comparison of the Rational Numbers with the Points of a Straight Line 6 III. Continuity of the Straight Line 8 IV. Creation of Irrational Numbers . . 12 V. Continuity of the Domain of Real Numbers ig VI. Operations with Real Numbers .... . 21 VII. Infinitesimal Analysis ... 24 II. THE NATURE AND MEANING OF NUMBERS. Prefaces . . 31 I. Systems of Elements . 44 II. Transformation of a System 50 III. Similarity of a Transformation. Similar Systems 53 IV. Transformation of a System in Itself 56 V. The Finite and Infinite 63 VI. Simply Infinite Systems. Series of Natural Numbers 67 VII. Greater and Less Numbers . , 70 VIII. Finite and Infinite Parts of the Number-Series 81 IX. Definition of a Transformation of the Number-Series by Induction . . . 83 X. The Class of Simply Infinite Systems . . 92 XI. Addition of Numbers . . 96 XII. Multiplication of Numbers . loi XIII. Involution of Numbers . . . 104 XIV. Number of the Elements of a Finite System . . 105 CONTINUITY AND IRRATIONAL NUMBERS CONTINUITY AND IRRATIONAL NUMBERS. TV /fY attention was first directed toward the consid- erations which form the subject of this pam- phlet in the autumn of 1858. As professor in the Polytechnic School in Ziirich I found myself for the first time obliged to lecture upon the elements of the differential calculus and felt more keenly than ever before the lack of a really scientific foundation for arithmetic. In discussing the notion of the approach of a variable magnitude to a fixed limiting value, and especially in proving the theorem that every magnitude which grows continually, but not beyond all limits, must certainly approach a limiting value, I had re- course to geometric evidences. Even now such resort to geometric intuition in a first presentation of the differential calculus, I regard as exceedingly useful, from the didactic standpoint, and indeed indispens- able, if one does not wish to lose too much time. But that this form of introduction into the differential cal- culus can make no claim to being scientific, no one will deny. For myself this feeling of dissatisfaction Was so overpowering that I made the fixed resolve to keep meditating on the question till I should find a 2 CONTINUITY AND purely arithmetic and perfectly rigorous foundation for the principles of infinitesimal analysis. The state- ment is so frequently made that the differential cal- culus deals with continuous magnitude, and yet an V explanation of this continuity is nowhere given ; even the most rigorous expositions of the differential cal- culus do not base their proofs upon continuity but, with more or less consciousness of the fact, they either appeal to geometric notions or those suggested by geometry, or depend upon theorems which are never established in a purely arithmetic manner. Among these, for example, belongs the above-men- tioned theorem, and a more careful investigation con- vinced me that this theorem, or any one equivalent to it, can be regarded in some way as a sufficient basis for infinitesimal analysis. It then only remained to discover its true origin in the elements of arithmetic and thus at the same time to secure a real definition of the essence of continuity. I succeeded Nov. 24, 1858, and a few days afterward I communicated the results of my meditations to my dear friend Durfege with whom I had a long and lively discussion. Later I explained these views of a scientific basis of arith- metic to a few of my pupils, and here in Braun- schweig read a paper upon the subject before the sci- entific club of professors, but I could not make up my mind to its publication, because., in the first place, the presentation did not seem altogether simple, and further, the theory itself had little promise. Never- IRRATIONAL NUMBERS. 3 theless I had already half determined to select this theme as subject for this occasion, when a few days ago, .March 14, by the kindness of the author, the paper Die Elemente der Funktionenlehre by E. Heine {Crelle's Journal, Vol. 74) came into my hands and confirmed me in my decision. In the main I fully agree with the substance of this memoir, and in- deed I could hardly do otherwise, but I will frankly acknowledge that my own presentation seems to me to be simpler in form and to bring out the vital point more clearly. While writing this preface (March 20, 1872), I am just in receipt of the interesting paper Ueber die Ausdehnung eines Satzes aus der Theorie.der trigonometrischen Reihen, by G. Cantor {Math. Annalen, Vol. 5),' for which I owe the ingenious author my hearty thanks. As I find on a hasty perusal, the ax- iom given in Section II. of that paper, aside from the form of presentation, agrees with what I designate in Section III. as the essence of continuity. But what advantage will be gained by even a purely abstract definition of real numbers of a higher type, I am as yet unable to see, conceiving as I do of the domain of real numbers as complete in itself. I. PROPERTIES OF RATIONAL NUMBERS. The development of the arithmetic of rational numbers is here presupposed, but still I think it worth while to call attention to certain important 4 CONTINUITY AND matters without discussion, so as to show at the out- set the standpoint assumed in what follows. I regard the whole of arithmetic as a necessary, or at least nat- ural, consequence of the simplest arithmetic act, that of counting, and counting itself as nothing else than the successive creation of the infinite series of positive integers in which each individual is defined by the one immediately preceding; the simplest act is the passing from an already-formed individual to the con- secutive new one to be formed. The chain of these numbers forms in itself an exceedingly useful instru- ment for the human mind; it presents an inexhaustible wealth of remarkable laws obtained by the introduc- tion of the four fundamental operations of arithmetic. Addition is the combination of any arbitrary repeti- tions of the above-mentioned simplest act into a sin- gle act ; from it in a similar way arises multiplication. While the performance of these two operations is al- ways possible, that of the inverse operations, subtrac- tion and division, proves to be limited. Whatever the immediate occasion may have been, whatever com- parisons or analogies with experience, or intuition, may have led thereto ; it is certainly true that just this limitation in performing the indirect operations has in each case been the real motive for a new crea- tive act ; thus negative and fractional numbers have been created by the human mind ; and in the system of all rational numbers there has been gained an in- strument of infinitely greater perfection. This system, IRRATIONAL NUMBERS. 5 which I shall denote by R, possesses first of all a com- pleteness and self-containedness which I have desig- nated in another place* as characteristic of a body of numbers [Zahlkarper] and which consists in this that the four fundamental operations are always perform- able with any two individuals in R, i. e., the result is always an individual of R, the single case of division by the number zero being excepted. For our immediate purpose, however, another property of the system R is still more important ; it may be expressed by saying that the system R forms a well-arranged domain of one dimension extending to infinity on two opposite sides. What is meant by this is sufficiently indicated by my use of expressions borrowed from geometric ideas ; but just for this rea- son it will be necessary to bring out clearly the corre- sponding purely arithmetic properties in order to avoid even the appearance as if arithmetic were in need of ideas foreign to it. To express that the symbols a and b represent one and the same rational number we put a = ^ as well as b:=:a. The fact that two rational numbers a, bare different appears in this that the difference a — b has either a positive or negative value. In the former case a is said to be greater than b, b less than a ; this is also indicated by the symbols a'^ b, b ':> a, a<.l>. In regard to these two ways in which two numbers may differ the following laws will hold: I. If a>b, and l)>c, then a> c. Whenever a, c are two different (or unequal) numbers, and b is greater than the one and less than the other, we shall, without hesitation because of the suggestion of geo- metric ideas, express this briefly by saying : b lies be- tween the two numbers a, c. II. If a, c are two different numbers, there are in- finitely many different numbers lying between a, c. III. If a is any definite number, then all numbers of the system R fall into two classes, A\ and Ai, each of which contains infinitely many individuals ; the first class A\ comprises all numbers ax that are a; the number a itself may be assigned at pleasure to the first or second class, being respectively the greatest number of the first class or the least of the second. In every case the separation of the system R into the two classes A-^, A^ is such that every num- ber of the first class A-^ is less than every number of the second class A%. II. COMPARISON OF THE RATIONAL NUMBERS WITH THE POINTS OF A STRAIGHT LINE. The above-mentioned properties of rational num- bers recall the corresponding relations of position of JKKAIIUJSIAL JSIUMBKKS. 7 the points of a straight line L. If the two opposite directions existing upon it are distinguished by "right" and "left," and/, q are two different points, then either / lies to the right of q, and at the same time q to the left of/, or conversely q lies to the right of/ and at the same time/ to the left of q. A third case is impossible, if p, q are actually different points. In regard to this difference in position the following laws hold : I. If / lies to the right of q, and q to the right of r, then / lies to the right of r; and we say that q lies between the points / and r. II. If /, r are two different points, then there al- ways exist infinitely many points that lie between p and r. III. If / is a definite point in L, then all points in L fall into two classes, P\, P^, each of which contains infinitely many individuals ; the first class Px contains all the points /i, that lie to the left of/, and the sec- ond class Pt contains all the points /2 that lie to the right of/ ; the point / itself may be assigned at pleas- ure to the first or second class. In every case the separation of the straight line L into the two classes or portions P\, Pi, is of such a character that every point of the first class Pi lies to the left of every point of the second class P^. This analogy between rational numbers and the points of a straight line, as is well known, becomes a real correspondence when we select upon the straight 8 CONTINUITY AND line a definite origin or zero-point o and a definite unit of length for the measurement of segments. With the aid of the" latter to every rational number a a cor- responding length can be constructed and if we lay this off upon the straight line to the right or left of o according as a is positive or negative, we obtain a definite end-point /, which may be regarded as the point corresponding to the number a ; to the rational number zero corresponds the point o. In this way to every rational number a, i. e., to every individual in R, corresponds one and only one point /, i. e. , an in- dividual in Z. To the two numbers a, b respectively correspond the two points /, q, and if a'^b, then p lies to the right of q. To the laws i, ii, iii of the pre- vious Section correspond completely the laws i, ii, iii of the present. III. CONTINUITY OF THE STRAIGHT LINE. Of the greatest importance, however, is the fact that in the straight line L there are infinitely many points which correspond to no rational number. If the point / corresponds to the rational number a, then, as is well known, the length op is commensur- able with the invariable unit of measure used in the construction, i. e., there exists a third length, a so- called common measure, of which these two lengths are integral multiples. But the ancient Greeks already IRRA'llONAL NUMBERS. q knew and had demonstrated that there are lengths in- commensurable with a given unit of length, e. g., the diagonal of the square whose side is the unit of length. If we lay off such a length from the point o upon the line we obtain an end-point which corresponds to no rational number. Since further it can be easily shown that there are infinitely many lengths which are in- commensurable with the unit of length, we may affirm: The straight line L is infinitely richer in point-indi- viduals than the domain R of rational numbers in number- individuals. If now, as is our desire, we try to follow up arith- metically all phenomena in the straight line, the do- main of rational numbers is insufficient and it becomes absolutely necessary that the instrument R constructed by the creation of the rational numbers be essentially improved by the creation of new numbers such that the domain of numbers shall gain the same complete- ness, or as we may say at once, the same continuity, as the straight line. The previous considerations are so familiar and well known to all that many will regard their repeti- tion quite superfluous. Still I regarded this recapitu- lation as necessary to prepare properly for the main question. For, the way in which the irrational num- bers are usually introduced is based directly upon the conception of extensive magnitudes-^which itself is nowhere carefully defined — and explains number as the result of measuring such a magnitude by another lo CONTINUITY AND of the same kind.* Instead of this I demand that arithmetic shall be developed out of itself. That such comparisons with non-arithmetic no- tions have furnished the immediate occasion for the ex- tension of the number-concept may, in a general way, be granted (though this was certainly not the case in the introduction of complex numbers); but this surely is no sufficient ground for introducing these foreign notions into arithmetic, the science of numbers. Just as negative and fractional rational numbers are formed by a new creation, and as the laws of operating with these numbers must and can be reduced to the laws of operating with positive integers, so we must en- deavor completely to define irrational numbers by means of the rational numbers alone. The question only remains how to do this. The above comparison of the domain R of rational numbers with a straight line has led to the recognition of the existence of gaps, of a certain incompleteness or discontinuity of the former, while we ascribe to the straight line completeness, absence of gaps, or con- tinuity. In what then does this continuity consist? Everything must depend on the answer to this ques- tion, and only through it shall we obtain a scientific basis for the investigation of all continuous domains. By vague remarks upon the unbroken connection in *The apparent advantage of the generality of this definition of number disappears as soon as we. consider complex numbers. According to my view, on the other hand, the notion of the ratio between two numbers of the same kind can be clearly developed only after the introduction of irrational num- bers. IRRATIONAL NUMBERS. ii the smallest parts obviously nothing is gained ; the problem is to indicate a precise characteristic of con- tinuity that can serve as the basis for valid deductions. For a long time I pondered over this in vain, but finally I found what I was seeking. This discovery will, perhaps, be differently estimated by different people ; the majority may find its substance very com- monplace. It consists of the following. In the pre- ceding section attention was called to the fact that every point p of the straight line produces a separa- tion of the same into two portions such that every point of one portion lies to the left of every point of the other. I find the essence of continuity in the con- verse, i. e., in the following principle : " If all points of the straight line fall into two classes such that every point of the first class lies to the left of every point of the second class, then there exists one and only one point which produces this di- vision of all points into two classes, this severing of the straight line into two portions." As already said I think I shall not err in assuming that every one will at once grant the truth of this statement; the majority of my readers will be very much disappointed in learning that by this common- place remark the secret of continuity is to be revealed. To this I may say that I am glad if every one finds the above principle so obvious and so in harmony with his own ideas of a line ; for I am utterly unable to adduce any proof of its correctness, nor has any 12 CONTINUITY AND one the power. The assumption of this property of the line is nothing else than an axiom by which we attribute to the line its continuity, by which we find continuity in the line. If space has at all a real ex- istence it is not necessary for it to be continuous ; many of its properties would remain the same even were it discontinuous. And if we knew for certain that space was discontinuous there would be nothing to prevent us, in case we so desired, from filling up its gaps, in thought, and thus making it continuous ; this filling up would consist in a creation of new point- individuals and would have to be effected in accord- ance with the above principle. IV. CREATION OF IRRATIONAL NUMBERS. From the last remarks it is sufficiently obvious how the discontinuous domain R of rational numbers may be rendered complete so as to form a continuous domain. In Section I it was pointed out that every rational number a effects a separation of the system R into two classes such that every number a\ of the first class A\ is less than every number a^ of the second ' ' class Ai ; the number a is either the greatest number of the class Ax or the least number of the class A^. If now any separation of- the system R into two classes A\, A3, is given which possesses only tkis characteris- tic property that every number ai in Ai is less than every number a^ in A^, then for brevity we shall call IRRATIONAL NUMBERS. 13 such a separation a cut [Schnitt] and designate it by (■^i, A<(\+1)2. If we assign to the second class A^, every positive rational number «2 whose square is >> D, to the first class Ax all other rational numbers a\, this separation forms a cut {A\, Ai), i. e., every number a\ is less than every number aj. For if ai = 0, or is negative, then on that ground a\ is less than any number ai, because, by definition, this last is positive ; if ax is positive, then is its square <2?, and hence ax is less than any positive number a^ whose square is >Z'. But this cut is produced by no rational number. To- demonstrate this it must be shown first of all that there exists no rational number whose square r=D. 14 CONTINUITY AND Although this is known, from the first elements of the theory of numbers, still the following indirect proof may find place here. If there exist a rational number whose square =Z>, then there exist two positive in- tegers t, u, that satisfy the equation and we may assume that u is the least positive integer possessing the property that its square, by multipli- cation by Z>, may be converted into the square of an integer /. Since evidently Xw) (/2 — Z»«2)=0, which is contrary to the assumption respecting u. Hence the square of every rational number x is either or >Z). From this it easily follows that there is neither in the class A-^ a greatest, nor in the class Ai a least number. For if we put x(^^ + 3Z>) ^ 3^2 + 2? ' we have and V.2 IRRATIONAL NUMBERS. 15 If in this we assume jc to be a positive number from the class A-i, then x^ -^D, and hence y^x and y^ 0, and y^^D. Therefore y likewise belongs to the class A^. This cut is therefore produced by no rational number. In this property that not all cuts are produced by rational numbers consists the incompleteness or dis- continuity of the domain R of all rational'numbers. Whenever, then, we have to do with a cut {Ai, A2) produced by no rational number, we create a new, an irrational number o, which we regard as completely defined by this cut {A\, A2); we shall say that the number a corresponds to this cut, or that it produces this cut. Frorn now on, therefore, to every definite cut there corresponds a definite rational or irrational number, and we regard two numbers as different or unequal always and only when they correspond to es- sentially different cuts. In order to obtain a basis for the orderly arrange- ment of all real, i. e., of all rational and irrational numbers we must investigate the relation between any two cuts {Ai, A2) and (^1, -£2) produced by any two numbers a and ^. Obviously a cut {Ai, A2) is given completely when one of the two classes, e. g., the first Ai is known, because the second A2 consists of all rational numbers not contained in Ai, and the characteristic property of such a first class lies in this i6 CONTINUITY AND that if the number a\ is contained in it, it also con- tains all numbers less than a\. If now we compare two such first classes A\, B\ with each other, it may happen 1. That they are perfectly identical, i. e., that every number contained in A\ is also contained in B\, and that every number contained in B\ is also contained in A\. In this case A^ is necessarily identical with B^, and the two cuts are perfectly identical, which we denote in symbols by a=j8 or ^ = 0,. But if the two classes Ai, Bi are not identical, then there exists in the one, e. g. , in Ai, a number a'i = d'i not contained in the other Bi and conse- quently found in B2 ; hence all numbers di contained in Bi are certainly less than this number a'i=3'2 and therefore all numbers 61 are contained in Ai. 2. If now this number a'l is the only one in Ai that is not contained in Bi, then is every other number ai contained in Ai also contained in Bi and is conse- quently j8 as well as j8a, a<;/8. As this exhausts the possible cases, it follows that of two different numbers one is necessarily the greater, the other the less, which gives two possibilities. A third case is impossible. This was indeed involved in the use of the comparative (greater, less) to desig- i8 CONTINUITY AND nate the relation between a, j8 ; but this use has only now been justified. In just such investigations one needs to exercise the greatest care so that even with the best intention to be honest he shall not, through a hasty choice of expressions borrowed from other no- tions already developed, allow himself to be led into the use of inadmissible transfers from one domain to the other. If now we consider again somewhat carefully the case a>/8 it is obvious that the less number j8, if rational, certainly belongs to the class Ax ; for since there is in Ai a number a'\=^b'i which belongs to the class Bi, it follows that the number /8, whether the greatest number in B\ or the least in B^, is certainly <«'i and hence contained in Ax. Likewise it is ob- vious from o> /3 that the greater number a, if rational, certainly belongs to the class B^,, because a> a'l. Com- bining these two considerations we get the following result : If a cut is produced by the number a then any rational number belongs to the class A\ or to the class At according as it is less or greater than' a; if the number a is itself rational it may belong to either class. From this we obtain finally the following : If o> /3, i. e., if there are infinitely many numbers in Ax not contained in Bx then there are infinitely many such numbers that at the same time are different from o and from j3 ; every such rational number c is < a, because IRRATIONAL NUMBERS. 19 it is contained in A\ and at the same time it is >/3 because contained in B^. V. CONTINUITY OF THE DOMAIN OF REAL NUMBERS. In consequence of the distinctions just estabhshed the system "S. of all real numbers forms a well-arranged domain of one dimension ; this is to mean merely that the following laws prevail : ^ I. If a>j8, and /3>y, then is also a>y. We shall SEy that the number ^ lies between u, and y. 1 II. If a, y are any two different numbers, then there exist infinitely many different numbers /3 lying between a, y. III. If a is any definite number then all numbers of the system H fall into two classes 2ti and "iXi each of which contains infinitely many individuals; the first class Hi comprises all the numbers ai that are less than a, the second JJa comprises all the numbers as that are greater than a ; the number a itself may be assigned at pleasure to the first class or to the second, and it is respectively the greatest of the first or the least of the second class. In each case the separation of the system H into the two classes Zli, "iXi is such that every number of the first class 2ti is smaller than every number of the second class 2t2 and we say that this separation is produced by the number a. For brevity and in order not to weary the reader I suppress the proofs of these theorems which follow ao CONTINUITY AND immediately from the definitions of the previous sec- tion. Beside these properties, however, the domain 21 possesses also continuity; i. e., the following theorem is true : IV. If the system "S. of all real numbers breaks up into two classes 2ti, Jfa such that every number a\ of the class 2ii is less than every number 02 of the class "iXi then there exists one and only one number a by which this separation is produced. Proof. By the separation or the cut of JJ into 2ti and 2I2 we obtain at the same time a cut {A\, Ai) of the system R of all rational numbers which is de- fined by this that Ax contains all rational numbers of the class 2Ii and Ai all other rational numbers, i. e., all rational numbers of the class "iXi- Let a be the perfectly definite number which produces this cut (y4i, A'i). If j8 is any number different from a, there are always infinitely many rational numbers c lying between a and ^. If ;8a, then is f>a; hence c belongs to the class A^ and consequently also to the class 2t2, and since at the same time y3>f, then ^ also belongs to the same class "iXi, because every number in "iXx is less than every number c in 212- Hence every number /3 differ- IRRAl^IONAL NUMBERS. 21 ent from a belongs to the class iti or to the class 21. according as ;8a; consequently o itself is either the greatest number in 2ti or the least number in 2l2, i. e., a is one and obviously the only number by which the separation of R into the classes 2ii, IXi is produced. Which was to be proved. VI. OPERATIONS WITH REAL NUMBERS. To reduce any operation with two real numbers a, ^ to operations with rational numbers, it is only necessary from the cuts {Ax, Ai), {Bx, B^ produced by the numbers a and /3 in the system R to define the cut (Ci, d) which is to correspond to the result of the operation, y. I confine myself here to the discus- sion of the simplest case, that of addition. If c is any rational number, we put it into the class C\, provided there are two numbers one a\ in A\ and one b\ in Bx such that their sum ax-\-bx~>c; all other rational numbers shall be put into the class^Ca. This separation of all rational numbers into the two classes C\, Ci evidently forms a cut, since every number ^i in Cx is less than every number fj in Cg. If both a and /8 are rational, then every number cx contained in Cx is a-\- p. Therefore in this case the cut (Ci, Cj) is produced by the sum 0+ j8. Thus we shall not violate the definition which holds in the arithmetic of rational numbers if in all cases we understand by the sum a-|-/3 of any two real numbers a, /3 that number y by which the cut (Ci, d) is produced. Further, if only one of the two numbers «,, /8 is rational, e. g., o, it is easy to see that it makes no difference with the sum •y^a + /3 whether the number o is put into the class Ai or into the class Ai- Just as addition is defined, so can the other ope- rations of the so-called elementary arithmetic be de- fined, viz., the formation of differences, products, quotients, powers, roots, logarithms, and in this way we arrive at real proofs of theorems (as, e. g., l/2 -l/S = 1/6), which to the best of my knowledge have never been established before. The excessive length that is to be feared in the definitions of the more complicated operations is partly inherent in the nature of the subject but can for the most part be avoided. Very useful in this connection is the notion of an interval, i. e., a system A of rational numbers possessing the follow- ing characteristic property: if a and a' are numbers of the system A, then are all rational numbers lying between a and a' contained in A. The system R of all rational numbers, and also the two classes of any IRRATIONAL NUMBERS. 23 cut are intervals. If there exist a rational number a\ which is less and a rational number ^2 which is greater than every number of the interval A, then A is called a finite interval ; there then exist infinitely many num- bers in the same condition as ci and infinitely many in the same condition as ai ; the whole domain R breaks up into three. parts A\, A, At and there enter two per- fectl}- definite rational or irrational numbers ai, 02 which may be called respectively the lower and upper (or the less and greater) limits of the interval; the lower limit ai is determined by the cut for which the system A\ forms the first class and the upper 02 by the cut for which the system ^2 forms the second class. Of every rational or irrational number a lying between ai and a2 it may be said that it lies within the interval A. If all numbers of an interval A are also numbers of an interval B, then A is called a portion of B.^, Still lengthier considerations seem to loom up when we attempt to adapt the numerous theorems of the arithmetic of rational numbers (as, e. g., the theo- rem {a -\- b^c ^ ac -{■ bc^ to, any real numbers. This, however, is not the case. It is easy to see that it all reduces to showing that the arithmetic operations possess a certain continuity. What I mean by this statement may be expressed in the form of a general theorem : " If the number A. is the result of an operation per- formed on the numbers a, )8, y, . . . and A. lies within the interval Z, then intervals A, B, C, . . . can be 24 CONTINUITY AND taken within which lie the numbers a, P, y, . . . such that the resujt of the same operation in which the numbers o, /?, y, . . . are replaced by arbitrary num- bers of the intervals A, B, C, . . . is always a number lying within the interval Z." The forbidding clumsi- ness, however, which marks the statement of such a theorem convinces us that something must be brought in as an aid to expression ; this is, in fact, attained in the most satisfactory way by introducing the ideas of variable magnitudes, functions, limiting values, and it would be best to base the definitions of even the sim- plest arithmetic operations upon these ideas, a matter which, however, cannot be carried further here. VII. INFINITESIMAL ANALYSIS. Here at the close we ought to explain the connec- tion between the preceding investigations and certain fundamental theorems of infinitesimal analysis. We say that a variable magnitude x which passes through successive definite numerical values ap- proaches a fixed limiting value a when in the course of the process x lies finally between two numbers be- tween which a itself lies, or, what amounts to the same, when the difference x^o. taken absolutely be- comes finally less than any given value different from zero. One of the most important theorems may be stated in the following manner : " If a magnitude ;*: grows IRRATIONAL NUMBERS. 25 continually but not beyond all limits it approaches a limiting value." I prove it in the following way. By hypothesis there exists one and hence there exist infinitely many numbers 02 such that x remains continually 0), then by hypothesis a time will come after which x will change by less than 8, i. e., if at this time x has the value a, then afterwards we shall continually have x^a — 8 and x as- Since every number ai is less than every number 02 there exists a perfectly definite num- ber a which produces this cut (2ti, 2X2) of the system K and which I will call the upper limit of the variable X which always remains finite. Likewise as a result of the behavior of the variable x a second cut (Bi, B2) of the system X is produced ; a number ySa (e.g., a — 8) is assigned to B2 when in the course of the pro- cess « becomes finally >|8; every other number /Sj, to be assigned to B2, has the property that x is never IRRATIONAL NUMBERS. 27 finally > ^82 ; therefore infinitely many times x becomes >)8 — £, but never finally^ j8-|-£. Now two cases are possible. If u and /8 are different from each other, then necessarily a>/3, since continually 02>/32; the variable x oscillates, and, however far the process advances, always under- goes changes whose amount surpasses the value (a — )8) — 2t where t is an arbitrarily small positive magnitude. The original hypothesis to which I now return contradicts this consequence ; there remains only the second case o = ;8 and since it has already been shown that, however small be the positive magni- tude e, we always have finally j£;;8 — e, JT approaches the limiting -value a, which was to be proved. These examples may suffice to bring out the con- nection between the principle of continuity and in- finitesimal analysis. THE NATURE AND MEANING OF NUMBERS PREFACE TO THE FIRST EDITION. TN science nothing capable of proof ought to be ac- ^ cepted without proof. Though this demand seems so reasonable yet I cannot regard it as having been met even in the most recent methods of laying the foundations of the simplest science; viz., that part of logic which deals with the theory of numbers.* In speaking of arithmetic (algebra, analysis) as a part of logic I mean to imply that I consider the number- concept entirely independent of the notions or intui- tions of space and time, that I consider it an imme- diate result from the laws of thought. My answer to the problems propounded in the title of this paper is, then, briefly this : numbers are free creations of the human mind; they serve. as a means of apprehending more easily and more sharply the difference of things. It is only through the purely logical process of build- ing up the science of numbers and by thus acquiring *Of the works which have come under my observatioi^ I mention the val- uable Lehrbuch der Arithntetik und Algebra of E. Schroder (Leipzig, 1873), which contains a bibliography of the subject, and in addition the niemoirs of Kronecker and von Helmholtz upon the Number-Concept and upon Counting and Measuring (in the collection of philosophical essays published in honor of E. Zeller, Leipzig, 1887). The appearance of these memoirs has induced me to publish my own views, in many respects similar but in foundation essentially different, which I formulated many years ago in absolute inde- pendence of the works of others. 32 THE NATURE AND the. continuous number-domain that we are prepared accurately to investigate our notions of space and time by bringing them into relation with this number- domain created in our mind.* If we scrutinise closely what is done in counting 'an aggregate or number of things, we are led to consider the ability of the mind to relate things to things, to let a thing corre- spond to a thing, or to represent a thing by a thing, an ability without which no thinking is i possible. Upon this unique and therefore absolutely indispen- sable foundation, as I have already affirmed in an an- nouncement of this paper, f must, in my judgment, the whole science of numbers be established. The design of such a presentation I had formed before the publication of my paper on Continuity, but only after its appearance and with many interruptions occa- sioned by increased official duties and other necessary labors, was I able in the years 1872 to 1878 to commit to paper a first rough draft which several mathemati- cians examined and partially discussed with me. It bears the same title and contains, though not arranged in the best order, all the essential fundamental ideas of my present paper, in which they are more carefully elaborated. As such main points I mention here the sharp distinction between finite and infinite (64), the notion of the , number \Anzahl~\ of things (161), the * See Section III. of my memoir, Continuity and Irrational Numbers (Braunschweig, 1872), translated at pages 8 et seq. of the present volume. tDirichlet's Vorlesungen jiber Zahlentheorie , third edition, 1879, § 163, note on page 470. MEANING OF NUMBERS. 33 proof that the form of argument known as conjplete induction (or the inference from « to « + l) is really conclusive (59), (60), (80), and that therefore the definition by induction (or recursion) is determinate and consistent (126). This memoir can be understood by any one pos- sessing what is usually called good common sense ; no technical philosophic, or mathematical, knowledge is in the least degree required. But I feel conscious that many a reader will' scarcely recognise in the shadowy forms which I bring before him his numbers which all his life long have accompanied him as faith- ful and familiar friends ; he will be frightened by the long series of simple inferences corresponding to our step-by-step understanding, by the matter-of-fact dis- section of the chains of reasoning on which the laws of numbers depend, and will become impatient at being compelled to follow out proofs for truths which to his supposed inner consciousness seem at once evi- dent and certain. On the contrary in just this possi- bility of reducing such truths to others more simple, no matter how long and apparently artificial the series of inferences, I recognise a convincing proof that their possession or belief in them is never given by inner consciousness but is always gained only by a more or less complete repetition of the individual inferences. I like to compare this action of thought, so difficult to trace on account of the rapidity of its performance, with the action which an accomplished reader per- 34 THE NA TURE AND forms in reading ; this reading always remains a more or less complete repetition of the individual steps which the beginner has to take in his wearisome spelling-out ; a very small part of the same, and there- fore a very small effort or exertion of the mind, is suffi- cient for the practised reader to recognise the correct, true word, only with very great probability, to be surej for, as is well known, it occasionally happens that even the- most practised proof-reader allows a typographical error to escape him, i. e., reads falsely, a thing which would be impossible if the chain of thoughts associated with spelling were fully repeated. So from the time of birth, continually and in increas- ing measure we are led to relate things to things and thus to use that faculty of the mind on which the creation of numbers depends ; by this practice con- tinually occurring, though without definite purpose, in our earliest years and by the attending formation of judgments and chains of reasoning we acquire a store of real arithmetic truths to which our first teach- ers later refer as to something simple, self-evident, given in the inner consciousness ; and so it happens that many very complicated notions (as for example that of the number \Anzahl'\ of things) are errone- ously regarded as simple. In this sense which I wish to express by the word formed after a well-known saying dei o avOpayrros a.pL6fii,rjn^a, I hope that the follow- ing pages, as an attempt to establish the science of numbers upon a uniform foundation will find a gener- MEANING OF NUMBERS. 35 ous welcome and that other mathematicians will be led to reduce the long series of inferences to more moderate and attractive proportions. In accordance with the purpose of this memoir I restrict myself to the consideration of the series of so-called natural numbers. In what way the gradual extension of the number-concept, the creation of zero, negative, fractional, irrational and complex numbers are to . be accomplished by reduction to the earlier notions and that without any introduction of foreign conceptions (such as that of measurable mag- nitudes, which according to my view can attain per- fect clearness ooly through the science of numbers), this I have shown at least for irrational numbers in my former memoir on Continuity (1872); in a way wholly similar, as I have already shown in Section III. of that memoir,* may the other extensions be treated, and I propose sometime to present this whole subject in systematic form. From just this point of view it appears as something self-evident and not new that every theorem of algebra and higher analysis, no mat- ter how remote, can be expressed as a theorem about natural numbers, — a declaration I have heard repeat- edly from the lips of Dirichlet. But I see nothing meritorious — and this was just as far from Dirichlet's thought — in actually performing this wearisome cir- cumlocution and insisting on the use and recognition of no other than rational numbers. On the contrary, * Pages 8 et seq. of the present volume. 36 THE NATURE AND the greatest and most fruitful advances in mathematics and other sciences have invariably been made by the creation and introduction of new concepts, rendered necessary by the frequent recurrence of complex phe- nomena which could be controlled by the old notions only with difficulty. On this subject I gave a lecture before the philosophic faculty in the summer of 1854 on the occasion of my admission as privat-docent in Gottingen. The scope of this lecture met with the approval of Gauss ; but this is not the place to go into further detail. Instead of this I will use the opportunity to make some remarks relating to my earlier work, mentioned above, on Continuity and Irrational Numbers. The theory of irrational numbers there presented, wrought out in the fall of 1853, is based on the phenomenon (Section IV.)* occurring in the domain of rational numbers which I designate by the term cut \Schnitt'\ and which I was the first to investigate carefully; it culminates in the proof of the continuity of the new domain of real numbers (Section V., iv. ).f It appears to me to be somewhat simpler, I might say easier, than the two theories, different from it and from each other, which have been proposed by Weierstrass and G. Cantor, and which likewise are perfectly rigorous. It has since been adopted without essential modifica- tion by U. Dini in his Fondamenti per la teorica delle * Pages 12 et seq. of the present volume, t Page 20 of the present volume. MEANING OF NUMBERS. 37 funzioni divariabili reali (Pisa, 1878); but the fact that in the course of this exposition my name happens to be mentioned, not in the description of the purely arithmetic phenomenon of the cut but when the au- thor discusses the existence of a measurable quantity corresponding to the cut, might easily lead to the sup- position that my theory rests upon the cpnsideration of such quantities. Nothing could be further from the truth; rather have I in Section III.* of my paper advanced several reasons why I wholly reject the in- troduction of measurable quantities ; indeed, at the end of the paper I have pointed out with respect to their existence that for a great part of the science of space the continuity of its configurations is not even a necessary condition, quite aside from the fact that in works on geometry arithmetic is only casually men- tioned by name but is never clearly defined and there- fore cannot be employed in demonstrations. To ex- plain this matter more clearly I note the following example : If we select three non-collinear points A, B, C at pleasure, with the single limitation that the ratios of the distances AB, AC, BC are algebraic numbers, t and regard as existing in space only those points M, for which' the ratios of AM, BM, CM to AB are likewise algebraic numbers, then is the space made up of the points M, as is easy to see, everywhere dis- * Pages 8 et seq. of the present volume. tDirichlet's Vorksungen iiher Zahknthearie, § 159 of the second edition, § 160 of the third. 38 THE NATURE AND continuous; but in spite of this discontinuity, and de- spite the existence of gaps in this space, all construc- tions that occur in Euclid's Elements, can, so far as I can see, be just as accurately effected as in perfectly continuous space ; the discontinuity of this space would not be noticed in Euclid's science, would not be felt at all. If any one should say that we cannot conceive of space as anything else than continuous, I should venture to doubt it and to call attention to the fact that a far advanced, refined scientific training is demanded in order to perceive clearly the essence of continuity and to comprehend that besides rational quantitative relations, also irrational, and besides al- gebraic, also transcendental quantitative relations are conceivable. All the more beautiful it appears to me that without any notion of measurable quantities and simply by a finite system of simple thought-steps man can advance to the creation of the pure continuous number-domain ; and only by this means in my view is it possible for him to render the notion of continu- ous space clear and definite. The same theory of irrational numbers founded upon the phenomenon of the cut is set forth in the Introduction cL la thdorie des fonctions d'une variable by J. Tannery (Paris, 1886). If I rightly understand a passage in the preface to this work, the author has thought out his theory independently, that is, at a time when not only my paper, but Dini's Fondamenti mentioned in the same preface, was unknown to him. MEANING OF NUMBERS. 39 This agreement seems to me a gratifying proof that my conception conforms to the nature of the case, a fact recognised by other mathematicians, e. g., by Pasch in his Einleitung in die Differential- und Integral- rechnung Q^ei^zig, 18833. But I cannot quite agree with Tannery when he calls this theory the develop- ment of an idea due to J. Bertrand and contained in his Traits d'ariihmdtique, consisting in this that an ir- rational number is defined by the specification of all rational numbers that are less and all those that are greater than the number to be defined. As regards this statement which is repeated by Stolz — apparently without careful investigation — in the preface to the second part of his Vorlesungen iiber allgemeine Arith- ■metik (Leipzig, 1886), I venture to remark the follow- ing : That an irrational number is to be considered as fully defined by the specification just described, this conviction certainly long before the time of Ber- trand was the common property of all mathematicians who concerned themselves with the notion of the irrational. Just this manner of determining it is in the mind of every computer who calculates the ir- rational root of an equation by approximation, and if, as Bertrand does exclusively in his book, (the eighth edition, of the year 1885, lies before me,) one regards the irrational number as the ratio of two measur- able quantities, then is this manner of determining it already set forth in the clearest possible way in the celebrated definition which Euclid gives of the equal- 40 MEANING OF NUMBERS. ity of two ratios (^Elements, V. , 5) . This same most ancient conviction has been the source of my theory as well as that of Bertrand and many other more or less complete attempts to lay the foundations for the introduction of irrational numbers into arithmetic. But though one is so far in perfect agreement with Tannery, yet in an actual examination he cannot fail to observe that Bertrand's presentation, in which the phenomenon of the cut in its logical purity is not even mentioned, has no similarity whatever to mine, inasmuch as it resorts at once to the existence of a measurable quantity, a notion which for reasons men- tioned above I wholly reject. Aside from this fact this method of presentation seems also in the succeed- ing definitions and proofs, which are based on the postulate of this existence, to present gaps so essential that I still regard the statement made in my paper (Section VI. ),* that the theorem l/2 • v 3 = V^ has no- where yet been strictly demonstrated, as justified with respect to this work also, so excellent in many other regards and with which I was unacquainted at that time. R. Dedekind. Harzburg, October 5, 1887. * Pages 21 et seq. of this voluma. PREFACE TO THE SECOND EDITION. npHE present memoir soon after its appearance met -*- with both favorable and unfavorable criticisms ; indeed serious faults were charged against it. I have been unable to convince myself of the justice of these charges, and I now issue a new edition of the memoir, which for some time has been out of print, without change, adding only the following notes to the first preface. The property which I have employed as the defi nition of the infinite system had been pointed out be- fore the appearance of my paper by G. Cantor (^Ein Beitrag zur Mannigfaltigkeitslehre, Crelle's Journal, Vol. 84, 1878), as also by Bolzano {Paradoxien des Unend- lichen, § 20, 1851). But neither of these authors made the attempt to use this property for the definition of the infinite and upon this foundation to establish with rigorous logic the science of numbers, and just in this consists the, content of my wearisome labor which in all its essentials I had completed several years before the appearance of Cantor's memoir and at a time when the work of Bolzano was unknown to me even by name. For the benefit of those who are interested in and understand the difficulties of such an investi- 42 THE NATURE AND gation, I add the following remark. We can lay down an entirely different definition of the finite and infinite, which appears still_ simpler since the notion of sim- ilarity of transformation is not even assumed, viz. : "A system S is said to be finite when it may be so transformed in itself (36) that no proper part (6) of S is transformed in itself ; in the contrary case S is called an infinite system." Now let us attempt to erect our edifice upon this new foundation! We shall soon meet with serious difficulties, and I believe myself warranted in saying that the proof of the perfect agreement of this defini- tion with the former can be obtained only (and then easily) when we are permitted to assume the series of natural numbers as already developed and to make use of the final considerations in (131); and yet noth- ing is said of all these things in either the one defini- tion or the other! From this we can see how very great is the number of steps in thought needed for such a remodeling of a definition. About a year after the publication of my memoir I became acquainted with G. Frege's Grundlagen der Arithmetik, which had already appeared in the year 1884. However different the view of the essence of number adopted in that work is from my own, yet it contains, particularly from § 79 on, ppints of very close contact with my paper, especially with my defi- nition (44). The agreement, to be sure, is not easy to discover on account of the different form of expres- MEANING OF NUMBERS. 43 sion; but the positiveness with which the author speaks of the logical inference from n\.on-\-\ (page 93, below) shows plainly that here he stands upon the same ground with me. In the meantime E. Schroder's Vorlesungen iiber die Algebra der Logik has been almost completed (1890-1891). Upon the importance of this extremely suggestive work, to which I pay my highest tribute, it is impossible here to enter further; I will simply confess that in spite of the remark made on p. 253 of Part I., I have retained my somewhat clumsy symbols (8) and (17); they make no claim to be adopted generally but are intended simply to serve the purpose of this arithmetic paper to which in my view they are better adapted than sum and product symbols. R. Dedekind. HARZBtTRG, August 24, 1893. THE NATURE AND MEANING OF NUMBERS. I. SYSTEMS OF ELEMENTS. 1. In what follows I understand by thing every object of our thought. In order to be able easily to speak of things, we designate them by symbols, e. g., by letters, and we venture to speak briefly of the thing a or of a simply, when we mean the thing de- noted by a and not at all the letter a itself. A thing is completely determined by all that can be affirmed or thought concerning it. A thing a is the same as b (identical with F), and b the same as a, when all that can be thought concerning a can also be thought con- cerning b, and when all that is true of b can also be thought of a. That a and-^ are only symbols or names for one and the same thing is indicated by the nota- tion a = /5, and also hyb = a. If further b = c, that is, if c as well as a is a symbol for the thing denoted by b, then is also a = c. If the above coincidence of the thing denoted by a with the thing denoted by b does not exist, then are the things a, b said to be dif- ferent, a is another thing than b, b another thing than MEANING OF NUMBERS. 45 a ; there is some property belonging to the one that does not belong to the other. 2. It very frequently happens that different things, a, b, c, . . . for some reason can be considered from a common point of view, can be associated in the mind, and we say that they form a system S\ we call the things a, b, c, . . . elements of the system S, they are contained in S; conversely, .S consists of these elements. Such a system S (an aggregate, a mani- fold, a totality) as an object of our thought is like- wise a thing (1); it is completely determined when with respect to every thing it is determined whether it is an element of .S or not.* The system ^ is hence the same as the system T, in symbols S^=T, when every element of S is also element of T, and every element of T is also element of S. For uniformity of expression it is advantageous to include also the spe- cial case where a system S consists of a single (one and only one) element a, i. e., the thing a is element of S, but every thing different from a is not an ele- ment of S. On the other hand, we intend here for certain reasons wholly to exclude the empty system which contains no element at all, although for other ♦ In what manner this determination is brought about, and whether we know a way of deciding upon it, is a matter of indifference for all that follows; the general laws to be developed in no way depend upon it; they hold under all circumstances. I mention this expressly because Kroneoker not long ago ICrelle's Journal^ Vol. gg, pp. 334-336I has endeavored to impose certain limi- tations upon the free formation of concepts in mathematics which I do not believe to be justified; but there seems to be no call to enter upon this mat- ter with more detail until the distinguished mathematician shall have pub- lished his reasons for the necessity or merely the expediency of these limi- 46 THE NATURE AMU investigations it may be appropriate to imagine such a system. 3. Definition. A system A is said to be part of a system S when every element of A is also element of 5. Since this relation between a system A and a sys- tem S will occur continually in what follows, we shall express it briefly by the symbol A^S. The inverse symbol Si A, by which the same fact might be ex- pressed, for simplicity and clearness I shall wholly avoid, but for lack of a better word I shall sometimes say that S is whole of A, by which I mean to express that among the elements of S are found all the ele- ments of A. Since further every element J of a system ^S* by (2) can be itself regarded as a system, we can hereafter employ the notation s^S. 4. Theorem. A^A, by reason of (3). 5. Theorem. If ^3^ and .53^, then ^=:^. The proof follows from (3), (2). 6. Definition. A system A is said to be a proper \echter'\ part of S, when A is part of S, but different from >S. According to (5) then 5' is not a part of A, i. e., there is in ^ an element which is not an element of ^. 7. Theorem. If A^B and B^C, which may be denoted briefly by AiBiC, then is AiC, and A is certainly a proper part of C, if ^ is a proper part of .5 or if ^ is a proper part of C. The proof follows from (3), (6). 8. Definition. By the system compounded out of MEANING OF NUMBERS. 47 any systems A, B, C, . . . to be denoted by 2T£ [A, B, C, . . .) we mean that system whose elements are de- termined by the following prescription: a thing is considered as element of ZTT (^, ^, C, . . .) when and only when it is element of some one of the systems A, B, C, . . ., '\. &., when it is element of A, or B, or C, . . . We include also the case where only a single system A exists; then obviously 2Ti {A') = A. We observe further that the system HI i.-^, B, C, . . .) compounded out of ^, ^, C, . . . is carefully to be dis- tinguished from the system whose elements are the systems A, B, C, . . . themselves. 9. Theorem. The systems A, B, C, . . . are parts of m {A, B, C,. . .)• The proof follows from (8), (3). 10. Theorem. If A, B, C, . . . are parts of a sys- tem S, then is 2TT {A, B, C, . . .) i S. The proof follows from (8), (3). 11. Theorem. If F is part of one of the systems A, B, C, . . . then is P^KR {A, B, C, . . .). The proof follows from (9), (7) . 12. Theorem. If each of the systems Z', Q, ■ . . is part of one of the systems A, B, C, . . . then is m {P, Q,---) ^m {A, B, C . .). The proof follows from (11), (10). 13. Theorem. If A is compounded out of any of the systems P, Q, . . . then is A^iVd {P, Q, ■ ■ •)• Proof. For every element of A is by (8) element of one of the systems P, Q, . . ., consequently by (8) 48 ' THE NATURE AND also element oi'SCi {P, Q, ■ ■ ■'), whence the theorem follows by (3). 14. Theorem. If each of the systems A, B, C, . . . is compounded out of any of the systems P, Q, . . . then is m(^A, B, c, . .)im{B, Q, ■ ■ ■) The proof follows from (13), (10). 15. Theorem. If each of the systems B, Q, . . is part of one of the systems A, B, C, . . ., and if each of the latter is compounded out of any of the former, then is va (.B, Q, . . .) =m {A, B, c, . . .). The proof follows from (12), (14), (5). 16. Theorem. If A = m {B, Q) and B = m. {Q, B) then is m {A, B) = -m{B, B). Proof. For by the preceding theorem (15) m {A, B) as well as m {B, B) = m (^B, Q, B). 17. Definition. A thing g is said to be common element of the systems A, B, C, ■ . ., \i it is contained in each of these systems (that is in A and in B and in C . . . .). Likewise a system 7^ is said to be a com- mon part oi A, B, C, . . . when 7" is part of each of these systems; and hy the community \Gemeinheit'\ of the systems A, B, C, . . . we understand the perfectly determinate system ® {A, B, C, . . .') which consists of all the common elements ^ of ^, ^, C, . . . and MEANING OF NUMBERS. 49 hence is likewise a common part of those systems. We again include the case where only a single system A occurs; then @ (^) (is to be put) =iA. But the case may also occur that the systems A, B, C, . . . possess no common element at all, therefore no com- mon part, no community ; they are then called sys- tems without common part, and the symbol ® {A, B, C, . . .) is meaningless (compare the end of (2)) We shall however almost always in theorems con- cerning communities leave it to the reader to add in thought the condition of their existence and to dis- cover the proper interpretation of these theorems for the case of non-existence. 18. Theorem. Every common part of A, B, C, . . . is part of © {A, B, C, . . .). The proof follows from (17). 19. Theorem. Every part of ® (^, B, C, . . .) is common part of A, B, C, . . . The proof follows from (17), (7). 20. Theorem. If each of the systems A, B, C, . . . is whole (3) of one of the systems P, Q, . . . then is ® {F, Q, . . .) i & {A, B, C, . . .) Proof. For every element of © {F, Q, . . .) is common element oi F, Q, . . ., therefore also common element of ^, ^, C, . . ., which was to be proved. 50 THE NATURE AND II. TRANSFORMATION OF A SYSTEM. 21. Definition.* By a transformation \Abbildung'\ <^ of a system S we understand a law according to which to every determinate element .<■ of S there be- longs a determinate thing which is called the trans- form of s and denoted by ^{s); we say also that <^(j) corresponds to the element s, that ^{/) results or is produced from s by the transformation <^, that j is transformed into <^ (j) by the transformation ^. If now 7" is any part of S, then in the transformation <^ of ^ is likewise contained a determinate transformation of T, which for the sake of simplicity may be denoted by the same symbol ^ and consists in this that to every element t of the system T there corresponds the same transform <^(^), which t possesses as element of S; at the same time the system consisting of all transforms ^ (/) shall be called the transform of T and be denoted by (^), by which also the significance of '^{S') is defined. As an example of a transformation of a sys- tem we may regard the mere assignment of deter- minate symbols or names to its elements. The sim- plest transformation of a system is that by which each of its elements is transformed into itself ; it will be called the identical transformation of the system. For convenience, in the following theorems (22), (23), (24), which deal with an arbitrary transformation ^ of *See Dirichlet's Vorlesungen ubcr Zaklentheorie, 3d edition, 1879, § 163. MEANING OF NUMBERS. 51 an arbitrary system S, we shall denote the transforms of elements j and parts T respectively by / and T' ; in addition we agree that small and capital italics without accent shall always signify elements and parts of this system S. 22. Theorem.* If ^3^, then ^' ^ .5'. Proof. For every element of A' is the transform of an element contained in A, and therefore also in B. and is therefore element of B' , which was to be proved. 23. Theorem. The transform of IXi {A, B, C, . . .) is W. (A', B', C, . . .). Proof. If we denote the system ZVi {A, B, C, . . .) which by (10) is likewise part of ^S by M, then is every element of its transform Af' the transform m' of an element m of Af; since therefore by (8) m is also ele- ment of one of the systems A, B, C, . . . and conse- quently m' element of one of the systems A', B', C , . . ., and hence by (8) also element of 2TI {A', B', C, . . .), we have by (8) M'^m(^A',B', C, . . .). On the other hand, since A, B, C, . . . are by (9) parts of M, and hence A', B', C, . . . by (22) parts of M', we have by (10) m{.A', B', C, . . .)^M'. By combination with the above we have by (5) the theorem to be proved M'^m{.A', B', C, . . .)• * See theorem 27. 52 THE NA rURE AND 24. Theorem.* The transform of every common part of A, B, C, . ., and therefore that of the com- munity © (^, B, C, . .) is part of ® (^', B' , C , . . .). Proof. For by (22) it is common part of A', B', C, . . ., whence the theorem follows by (18). 25. Definition and theorem. If is a transforma- tion of a system S, and ^ a transformation of the transform ^' = <^ (5), there always results a transfor- mation 6 of S, compounded\ out of <^ and i/r, which con- sists of this that to every element s oi S there corres- ponds the transform e(4='A(0=='A(<^W). where again we have put (s)=s'. This transforma- tion can be denoted briefly by the symbol \f/.^ or {j/(f>, the transform ^(j-) by i/f(^(j') where the order of ihe symbols 4>, i/r is to be considered, since in general the symbol \l/ has no interpretation and actually has meaning only when tl/{s')is. If now x signifies a transformation of the system il/(s') = ij/\s) and rj the transformation x'P '^^ the system S' compounded out of f and X. then is x^(^) = X>/'(-f') = '?C0 ='?{a'), b'^^{b). Since in this case conversely from / = t' we always have s = t, then is every element of the system »S" = ^ (5') the transform s' of a single, perfectly determi- nate element s of the system S, and we can therefore set over against the transformation ^ of ,5 an inverse transformation of the system S', to be denoted by ^, which consists in this that to every element / of S' there corresponds the transform ^{s')=s, and obvi- ously this transformation is also similar. It is clear that ^(5") = .5', that further ^ is the inverse transformation belonging to ^ and that the transformation ^<^ com- pounded out of and $ by (25) is the identical trans- formation of ^ (21). At once we have the following additions to II., retaining the notation there given. 27. Theorem.* If ^' 3,5', then ^ 3.5. Proof. For if a is an element of A then is a' an element of A', therefore also of £', hence =^', where b is an element of B; but since from a' = b' we always * See theorem 22. 54 THE NATURE AND have a=^b, then is every element oi A also element of B, which was to be proved. 28. Theorem. If ^'=^', then ^ =^. The proof follows from (27), (4), (5). 29. Theorem.* If G = &{A, B, C, . . .), then G' = &{A',B', C, . . .). Proof. Every element of &{A', B', C, . . .) is certainly contained in S', and is therefore the trans- form ^' of an element ^ contained in ^; but since ^' is common element of A', B', C, . . . then by (27) must g be common element of A, B, C, . . . therefore also element of G; hence every element of (S(^', B', C", . . .) is transform of an element g of G, therefore element of G', i. e., ®{A', B', C, . . .)iG', and ac- cordingly our theorem follows from (24), (5). 30. Theorem. The identical transformation of a system is always a similar transformation. 31. Theorem. If <^ is a similar transformation of S and i/f a similar transformation of (6'), then is the transformation i//^ of S, compounded of and ip, a sim- ilar transformation, and the associated inverse trans- formation ij/{b) and therefore ip is a. similar transfor- mation. Besides every element i/r<^(j-)z=i^(/) of the system i/f<^ (5) is transformed by ^ into / = <^(j-) and * See theorem 24. MEANING OF NUMBERS. 55 this by ^ into s, therefore ^^(j') is transformed by (S)=^ Q, which was to be proved. 34. Definition. We can therefore separate all sys- tems into classes by putting into a determinate class all systems Q, R, S, . . ., and only those, that are similar to a determinate system R, the representative of the class; according to (33) the class is not changed by taking as representative any other system belong- ing to it. 35. Theorem. If R, S are similar systems, then is every part of S also similar to a part of ^, every proper part of S also similar to a proper part of R. Proof. For if ^ is a similar transformation of S, (S) — R, and TiS, then by (22) is the system sim- ilar to T{T)iR; if further T is proper part of S, and s an element of S not contained in T, then by (27) the element ^(i') contained in R cannot be contained in {T); hence in Z. Hence we call <^ a transformation of the system S in itself, when <^(6')36', and we propose in this paragraph to investi- gate the general laws of such a transforrnation <^. In doing this we shall use the same notations as in II. and again put <^(j-) = j', (T)=T'. These trans- forms s', T' are by (22), (7) themselves again ele- ments or parts of S, like all things designated by italic letters. 37. Definition. ^ is called a chain \Keite\, when K'^K. We remark expressly that this name does not in itself belong to the part K of the system S, but is given only with respect to the particular transfor- mation <^ ; with reference to another transformation of the system S in itself K can very well not be^ chain. 38. Theorem. 6' is a chain. 39. Theorem. The transform K' of a chain Z'is a chain. Proof. For from K'^K it follows by (22) that i^K'^'^K', which was to be proved. 40. Theorem. If A is part of a chain K, then is also ^'3^. MEANING OF NUMBERS. 57 Proof. For from Ai K it follows by (22) that A'^K', and since by (37) K'lK, therefore by (7) A'^K, which was to be proved. 41. Theorem. If the transform A' is part of a chain L, then is there a chain K, which satisfies the conditions A^K, K'iL ; and 2.TT(^, Z) is just such a chain K. Proof. If we actually put K='iX\. {A, L), then by (9) the one condition Ai K is fulfilled. Since further by (23) K'^VTiiA', L') and by hypothesis A'iL, L'-iL, then by (10) is the other condition K'^L also fulfilled and hence it follows because by (9) LiK, that also K'iK, i. e. , ^ is a chain, which was to be proved. 42. Theorem. A system J^ compounded simply out of chains A, B, C, . . . \s 2. chain. Proof. Since by (23) M' = m{A', B', C',. ..) and by hypothesis ^'3^, B'iB, C'iC, . . . therefore by (12) M'iM, which was to be proved. 43. Theorem. The community G of chains A B, C, ... is a. chain. Proof. Since by (17) G is common part of A, B, C, . . ., therefore by (22) G' common part of A', B', C, . . ., and by hypothesis A'iA, B'iB, C'iC, . . ., then by (7) G' is also common part of A, B, C, . . . and hence by (18) also part of G, which was to be proved. 44. Definition. If A is any part of S, we will de- note by A^ the community of all those chains (e.g., S) 58 THE NATVRE AND of which A is part ; this community A„ exists (17) be- cause A is itself common part of all these chains. Since further by (43) A^ is a chain, we will call A„ the chain of the system A, or briefly the chain of A. This definition too is strictly related to the fundamen- tal determinate transformation ^ of the system S in itself, and if later, for the sake of clearness, it is necessary we shall at pleasure use the symbol 4'o{^) instead of A„ and likewise designate the chain of A corresponding to another transformation m by (o„{A). For this very important notion the following theorems hold true. 45. Theorem. AiA^. Proof. For A is common part of all those chains whose community is A^, whence the theorem follows by (18). 46. Theorem. {A„yiA^. Proof. For by (44) A„ is a chain (37). 47. Theorem. If A is part of a chain X, then is ailsoAJK. Proof. For A„ is the community and hence also a common part of all the chains JC, of which A is part. 48. Remark. One can easily convince himself that the notion of the chain A^ defined in (44) is com- pletely characterised by the preceding theorems, (45), (46), (47). 49. Theorem. A'i(AJ. The proof follows from (45), (22). MEANING OF NUMBERS. jg 50. Theorem. A'^A^. The proof follows from (49), (46), (7). 51. Theorem. If ^ is a chain, then A^^A. Proof. Since A is part of the chain A, then by (47) A„iA, whence the theorem follows by (45), (5). 52. Theorem. If ^3^, then ^3 ^„. The proof follows from (45), (7). 53. Theorem. If BiA„ then £JA„ and con- versely. Proof. Because A^ is a chain, then by (47) from £iA„, we also get BJA„; conversely, if ^„3^„, then by (7) we also get BiA^, because by (45) £iB„. 54. Theorem. If BiA, then is B^iA^. The proof follows from (52), (53). 55. Theorem. If BiA„ then is also B'iA^. Proof. For by (53) B„iA„, and since by (50) B'iB,, the theorem to be proved follows by (7). The same result, as is easily seen, can be obtained from (22), (46), (7), or also from (40). 56. Theorem. If BiA^, then is {B^)'i{A„)'. The proof follows from (53), (22). 57. Theorem and definition. {A^)' = {A\, i. e., the transform of the chain of A is at the same time the chain of the transform of A. Hence we can desig- nate this system in short by A'^ and at pleasure call it the chain-transform or transform-chain of A. With the clearer notation given in (44) the theorem might be expressed by <^(<^„(^)) = <^„(^(^)). Proof. If for brevity we put {A')^ = L, Z is a 6o THE NATURE AND chain (44) and by (45) A'^L; hence by (41) there ex- ists a chain ^satisfying the conditions A^K, K'iL; hence from (47) we have AJK, therefore {A^j'iX', and hence by (7) also {A^yiL, i. e., (^„)'3(^')„. Since further by (49) A'i{A,y, and by (44), (39) {A^' is a chain, then by (47) also (^')=3(^o)', whence the theorem follows by combining with the preceding result (5). 58. Theorem. ^„ = 2n(^, A\), i. e., the chain of A is compounded out of A and the transform-chain of ^. Proof. If for brevity we again put L=A', = {A„y^{A'\ and K=m{A, L), then by (45) A'iL, and since Z is a chain, by (41) the same thing is true of X; since further yi 3 X (9), therefore by (47) AJX. On the other hand, since bj' (45) AiA„ and by (46) also LIA„ then by (10) also £:iA„ whence the theorem to be proved A„^:^K follows by combining with the preceding result (5). 59. Theorem of complete induction. In order to show that the chain A^ is part of any system S — be this latter part of »S or not — it is sufficient to show, p. that ^35, and MEANING OF NUMBERS. 6i 0-. that the transform of every common element of Ag and S is likewise element of S. Proof. For if p is true, then by (45) the com- munity G^^{A„ 2) certainly exists, and by (18) A^G ; since besides by (17) G^A,, then is G also part of our system S, which by <^ is transformed in itself and at once by (55) we have also G'iA„. If then o- is likewise true, i. e., if G'^% then must G' as common part of the systems A„ 2 by (18) be part of their community G, i. e., G^ is a chain (37), and since, as above noted, A i G, then by (47) is also A^IG, and therefore by combination with the preceding re- sult G^A„, hence by (17) also A„i% which was to be proved. 60. The preceding theorem, as will be shown later, forms the scientific basis for the form of demonstra- tion known by the name of complete induction (the inference from n to n-\-V) ; it can also be stated in the following manner : In order to show that all ele- ments of the chain A„ possess a certain property (£ (or that a theorem S dealing with an undetermined thing n actually holds good for all elements n of the chain A^ it is sufficient to show p. that all elements a of the system A possess the property (£ (or that 5 holds for all a's) and a-, that to the transform n' of every such element n of A„ possessing the property (g, belongs the same 62 THE NATURE AND property (g (or that the theorem S, as soon as 'it holds for an element n of A„ certainly must also hold for its transform «'). Indeed, if we denote by S the system of all things possessing the property (£ (or for which the theorem S holds) the complete agreement of the present man- ner of stating the theorem with that employed in (59) is immediately obvious. 61. Theorem. The chain of KXii^A, B, C, . . .) is ZTK^o, B„ C, ). Proof. If we designate by M the former, by K the latter system, then by (42) X is a chain. Since then by (45) each of the systems A, B, C, . . . is part of one of the systems A„ B^, C„ . . ., and therefore by (12) MiK, then by (47) we also have On the other hand, since by (9) each of the systems A, B, C, . . , is part of M, and hence by (45), (7) also part of the chain M„ then by (47) must also each of the systems A„ B„ C„ . . . be part of M„ therefore by (10) whence by combination. with the preceding result fol- lows the theorem to be proved M„^X (5). 62. Theorem. The chain of &{A, B, C, . . .) is partof ®(^„, B„ C„ . . .). Proof. If we designate by G the former, by Xthe latter system, then by (43) X is a chain. Since then each of the systems A^, B„ C„, . . . by (45) is whole MEAmNG OF NUMBERS. 63 of one of the systems A, B, C, . . ., and hence by (20) GiX, therefore by (47) we obtain the theorem to be proved G„iX. 63. Theorem. If X'iLiX, and therefore X is a chain, Z is also a chain. If the same is proper part of X, and U the system of all those elements of 75' which are not contained in Z, and if further the chain U'o'is proper part of JC, and Fthe system of all those elements of X which are not contained in [/„, then is X^mi^o, y)&ndZ = -m{b')^\p{b), because i/f is a similar transfor- mation ; if further a is contained in T, but b not, then is ^ (a!) = i/> (a) different from (j>[b)^b, because i/'(«) is contained in T; if finally neither a nor b is con- tained in 7^ then also is 4)(a) = a different from (^{b^^b, which was to be shown. Since further \p{T') is part of T, because by (7) also part of S, it is clear that also (j)(S)iS. Since finally i/'(7') is proper part of 7' there exists in 7' and therefore also in S, an element /, not contained in i/'( 7") =<^(7') ; since then the transform 4>(/) of every element s not contained in 7" is equal to s, and hence is different from /, / cannot be contained in <^(^) ; hence <^('S') is proper part of .Sand conse- quently S is infinite, which was to be proved. 69. Theorem. Every system which is similar to a part of a finite system, is itself finite. The proof follows from (67), (68). 70. Theorem. If a is an element of S, and if the aggregate T of all the elements of S different from a is finite, then is also S finite. 66 THE NATURE AND Proof. We have by (64) to show that if <^ denotes any similar transformation of S in itself, the trans- form <^(^) or 5' is never a proper part of ^ but al- ways ^S. Obviously S^TXlifl, T) and hence by (23), ff the transforms are again denoted by accents, 5' = ItT («', T'^, and, on account of the similarity of the transformation , a' is not contained in T' (26). Since further by hypothesis S' -i S, then must a' and like- wise every element of T' either =a, or be element of 71 If then — a case which we will treat first — a is not contained in T', then must 7"3rand hence T'=T, because ^ is a similar transformation and because T\s a finite system; and since a', as remarked, is not con- tained in T', i.e., not in T, then must a' ^^a, and hence in this case we actually have 5" = 5 as was stated. In the opposite case when a is contained in T' and hence i& the transform b' of an element b contained in T, we will denote by U the aggregate of all those elements u of T, which are different from b ; then T='X!Ci{b,U) and by (] 5) ^=2n («, b, U), hence S' --=7X1 («', a, U'). We now determine a new transformation tf/ o{ T \n which we put tp{b)^a', and generally ^{ii) = u', whence by (23) yp{T)^m{a', U'). Obviously xjr is a similar transformation, because was such, and be- cause a is not contained in C/"and therefore also a' not in U'. Since further a and every element u is differ- ent from b then (on account of the similarity of ^) must also a' and every element u' be different from a and consequently contained in T; hence i/f(7')Br MEANING OF NUMBERS. 67 and since T is finite, therefore must \j/(^T) ^T, and rrXCa', U")=T. From this by (15) we obtain m(a', a, C^') = iTt(«, r) i. e., according to the preceding S' =S. Therefore in this case also the proof demanded has been se- cured. VI. SIMPLY INFINITE SYSTEMS. SERIES OF NATURAL NUMBERS. 71. Definition. A system N is said to be simply infinite when there exists a similar transformation <^ of N in itself such that iV appears as chain (44) of an element not contained in ^ (^)^ We call this ele- ment, which we shall denote in what follows by the symbol 1, the base-element of N and say the simply infinite system N is set in order \_geordnet'\ by this transformation <^. If we retain the earlier convenient symbols for transforms and chains (IV) then the es- sence of a simply infinite system N consists in the existence of a transformation <^ oi N and an element 1 which satisfy the following conditions a, ji, y, h: a. JV'iN. p. N=K y. The element 1 is "not contained in N'. S. The transformation <^ is similar. . Obviously it follows from a, y, 8 that every simply in- finite system Nis actually an infinite system (64) be- cause it is similar to a proper part N' of itself. 68 THE NATURE AND 72. Theorem. In every infinite system S a simply infinite system N is contained as a part. Proof. By (64) there exists a similar transforma- tion ^ oi S such that <^(5) or S' is a proper part of S\ hence there exists an element 1 in ^ which is not contained in S' . The chain iV=l„, which corresponds to this transformation <^ of the system S in itself (44), is a simply infinite system set in order by <^ ; for the characteristic conditions u,, /3, y, S in (71) are obvi- ously all fulfilled. 73. Definition. If in the consideration of a simply infinite system iV" set in order by a transformation (^ we entirely neglect the special character of the ele- ments; simply retaining their distinguishability and. taking into account only the relations to one another in which they are placed by the order- setting trans- formation <^, then are these elements called natural numbers or ordinal numbers or simply numbers, and the base-element 1 is cs\\&A\h% base-number oi'^e. number- series N. With reference to this freeing the elements from every other content (abstraction) we are justified in calling numbers a free creation of the human mind. The relations or laws which are derived entirely from the conditions a, j8, y, 8 in (71) and therefore are al- ways the same in all ordered simply infinite systems, whatever names may happen to be given to the indi- vidual elements (compare 134), form the first object of the science of numbers or arithi^ietic. From the general notions and theorems of IV. about the transformation MEANING OF NUMBERS. 69 of a system in itself we obtain immediately the follow- ing fundamental laws where a, b, . . in, n, . . always denote elements of N, therefore numbers, A, B, C, . . . parts of N, a', b' , . . . m' , n' , . . . A', B' , C . . . the corresponding transforms, which are produced by the order-setting transformation ^ and are always ele- ments or parts of N; the transform «' of a number n is also called the vmxah&T following n. 74. Theorem. Every number n by (45) is con- tained in its chain n^ and by (53) the condition rAm„ is equivalent to n„im„. 75. Theorem. By (57) ;/,= («„)'= («')»• 76. Theorem. By (46) «',3«,. 77. Theorem. By (58) n, = m{n, «'„). 78. Theorem. iV"= ITT (1 , jV') , hence every num- ber different from the base-number 1 is element of N', i. e., transform of a number. The proof follows from (77) and (71). 79. Theorem. iVis the only number-chain con- taining the base-number 1. Proof. For if 1 is element of a number-chain K, then by (47) the associated chain N^K, hence N=zK, because it is self-evident that K^N. 80. Theorem of complete induction (inference from n to «'). In order to show that a theorem holds for all numbers ;? of a chain ni„ it is sufficient to show, p. that it holds for n^=m, and 0-. that from the validity of the theor&m for a num- yo THE NATURE AND ber « of the chain m„ its validity for the following number ri always follows. This results immediately from the more general theorem (59) or (60). The most frequently occurring case is where m = 1 and therefore m„ is the complete number-series iV. VII. GREATER AND LESS NUMBERS. 81. Theorem. Every number n is different from the following number «'. Proof by complete induction (80) : p. The theorem is true for tlie number « = 1, be- cause it is not contained in JV' (71), while the follow- ing number 1' as transform of the number 1 contained in iVis element of JV. a-. If the theorem is true for a number n and we put the following number «'=/, then is n different from /, whence by (26) on account of the similarity (71) of the order-setting transformation <^ it follows that n', and therefore /, is different from /'. Hence the theorem holds also for the number/ following n, which was to be proved. 82. Theorem. In the transform-chain n\ of a num- ber n by (74), (75) is contained its transform n', but not the number n itself. Proof by complete induction (80) : p. The theorem is true for « = 1, because 1\=JV, MEANING OF NUMBERS. ^i and because by (71) the base-number 1 is not con- tained in N' . « (90) we use the symbols mm 74 THE NA TURB AND and we say m is at most equal to n, and n is at least equal to m. 93. Theorem. Each of the conditions m^n, nK^n', n^im, is equivalent to each of the others. Proof. For if ?nKn, then from X, fx. in (90) we always have n^^ni„ because by (76) m\-^m. Con- versely, if n^-im„ and therefore by (74) also nim„ it fol- lows from »z„=:ITT(»^ »z'„) that either n = m, or nim\, i. e., n'^jn. Hence the condition ?/iKn is equivalent to n^im^. Besides it follows from (22), (27), (75) that this condition n„^m„ is again equivalent to n\im\, i. e., by ;u. in (90) to ml„ and ■n^\m^ we have also «„3 /„, which was to be proved. 96. Theorem. In every part T oi N there exists one and only one least number k, i. e., a number k MEANING OF NUMBERS. 75 which is less than every other number contained in T. If 7" consists of a single number, then is it also the least number in T. Proof. Since T„ is a chain (44), then by (87) there exists one number k whose chain k„^ T„. Since from this it follows by (45), (77) that TiTXlik, k\), then first must k itself be contained in T (because other- wise T^k'„ hence by (47) also T„ik\, i. e., k^k\, which by (83) is impossible), and besides every num- ber of the system T, different from k, must be con- tained in k\, i. e., be >^ (89), whence at once from (90) it follows that there exists in T one and only one least number, which was to be proved. 97. Theorem. The least number of the chain n„ is n, and the base-number 1 is the least of all numbers. Proof. For by (74), (93) the condition m^n^ is equivalent to m>^n. Or our theorem also follows im- mediately from the proof of the preceding theorem, because if in that we assume T=^n^, evidently /J = « (51). 98. Definition. If n is any number, then will we denote by Z„ the system of all numbers that are not greater than n, and hence not contained in «'„. The condition m^Z„ by (92), (93) is obviously equivalent to each of the following conditions : mn, fKim, Z„3Z^. The proof follows immediately from (90) if we ob- serve that by (100) the conditions «„3 m„ and Z^iZ„ are equivalent. 102. Theorem. Zi = l. Proof. For by (99) the base-number 1 is con- tained in Zi, while by (78) every number different from 1 is contained in 1\, hence by (98) not in Zi, which was to be proved. 103. Theorem. By (98) JV=m(,Z„, n\). 104. Theorem. « = ®(Z„, n,), i. e., n is the only common element of the system Z„ and «„. Proof. From (99) and (74) it follows that n is MEANING OF NUMBERS. 77 contained in Z„ and n„ ; but every element of the chain n„ different from n by (77) is contained in «'„, and hence by (98) not in Z„, which was to be proved. 105. Theorem. By (91), (98) the number n' is not contained in Z„. 106. Theorem. If m /, a least number u, which and which alone possesses the properties stated in the theorem. In like manner the correctness of the last part of the theorem is obvious. 118. Theorem. In i\Athe number n' is next greater than n, and n next less than n' . The proof follows from (116), (117). MEANING OF NUMBERS. gi VIII. FINITE AND INFINITE PARTS OF THE NUMBER- SERIES. 119. Theorem. Every system Z„ in (98) is finite. Proof by complete induction (80). p. By (65), (102) the theorem is true for « = 1. 0-. If Z„ is finite, then from (108) and (70) it fol- lows that Z„. is also finite, which was to be proved. 120. Theorem. If m, n are different numbers, then are Z„, Z„ dissimilar systems. Proof. By reason of the symmetry we may by (90) assume that m«i, and since generally u<^ij/{u), then by (95) «i < <{i{u), and therefore by (90) ui is different from ijf(u), i. e., y. the element ai of (7 is not contained in xj/^C/). Therefore ij/ {17) is proper part of 17 and hence by (64) C/'is an infinite system. If then in agreement with (44) we denote by xj/^{V"), when F is any part of 17, the chain of F corresponding to the transformation f, we wish to show finally that ^. C/=i/.„(«i)- In fact, since every such chain i/r„ ( V) by reason of its definition (44) is a part of the system [7 transformed in itself by ij/, then evidently is \j/„ (ui ) 3 C/"; conversely it is first of all obvious from (45) that the element ui contained in C^is certainly contained in i/r„(«i); but if we assume that there exist elements of U, that are not contained in \^„{u{), then must there be among them by (96) a least number w, and since by what precedes this is different from the least number «i of the system 17, then by (117) must there exist in U also a number v which is next less than w, whence it MEANING OF NUMBERS. 83 follows at once that zc/=:c^(w); since therefore v<^w, then must v by reason of the definition of w certainly be contained in ^,{ui); but from this by (55) it fol- lows that also ^{v'), and hence w must be contained in ^„{u\), and since this is contrary to the definition of w, our foregoing hypothesis is inadmissible ; therefore C/iij/„{ui) and hence also f =i/'„(«i), as stated. From a, /3, y, 8 it then follows by (71) that C^is a simply in- finite system set in order by i[/, which was to be proved. 123. Theorem. In consequence of (121), (122) any part 7' of the number-series iV^is finite or simply infinite, according as a greatest number exists or does not exist in T. IX. DEFINITION OF A TRANSFORMATION OF THE NUMBER-SERIES BY INDUCTION. 124. In what follows we denote numbers by small Italics and retain throughout all symbols of the pre- vious sections VI. to VIII., while O designates an arbitrary system whose elements are not necessarily contained in JV. 125. Theorem. If there is given an arbitrary (sim- ilar or dissimilar) transformation oi a system O in itself, and besides a determinate element u in £i, then to every number n corresponds one transformation ■ ilf„ and one only of the associated number-system Z„ explained in (98), Vhich satisfies the conditions:* *For clearness here and in the following theorom (126) I have especially mentioned condition I., although properly it is a consequence oi II. and III 84 THE NATURE AND I. ^„(Z„)3n II. l/.„(l) = 0) III. !/'„('') = ^'/',X0> if t Iir. ij/j,(m') =^011/^(7/1), when mfc> («)=l!{n)=xl>„{n). («) Since thus \p is completely determined it follows also that there can exist only one such transformation \\i (see the closing remark in (130)). That conversely the transformation \f/ determined by («) also satisfies our conditions I, II, III, follows easily from («) with reference to the properties I, II and (/) shown in (125)j which was to be proved. 127. Theorem. Under the hypotheses made in the foregoing theorem, where T denotes any part of the number-series N. Proof. For if / denotes every number of the sys- tem T, then ip{T') consists of all elements i/f(^'), and 6\p{T) of all elements Oij/{^); hence our theorem fol- lows because by III in (126) \jf{t') — 6\ir(_t). 128. Theorem. If we~maintain the same hypoth- eses and denote by 0„ the chains (44) which corre- spond to the transformation 6 of the system O in itself, then is MEANING OF NUMBERS. 87 Proof. We show first by complete induction (80) that ./r(i\^)3e„(co), i. e., that every transform i//(«) is also element of e„(<«)- In fact, p. this theorem is true for n = \, because by (126, II) i^(l) = (<), and because by (45) (o3^„() is contained in \j/(JV), therefore that we likewise apply complete induction, i. e., theorem (59) transferred to CI and the transformation 6. In fact, p. the element , which may be denoted hy XA) defined in (i4i), where A again denotes any part of S. If for brevity we denote by A„ the transform m"(A) produced by the transformation a", then from III and (25) it follows that io(A„) = A„,. Hence it is easily shown by complete induction (80) that all these systems A„ are parts of the chain o)„(^) ; for p. by (50) this statement is true for «= 1, and a-, if it is true for a number n, then from (55) and from A„,z=w(A„) it follows that it is also true for the following number «', which was to be proved. Since 92 THE NATURE AND further by (45) ^3(o„(^), then from (10) it results that the system K compounded out of A and all transforms A„ is part of a),(^)- Conversely, since by (23) a)(i5r) is compounded out of o)(^) = ^i and all systems is a transformation of a system 6' in itself, and A any part of S, then is the chain of A corresponding to the trans- formation (1) compounded out of A and all the trans- forms n\, and the theorem to prove comes to this that \^{n) is not contained in (^n)=n', and to this again a deter- minate element tj/in") in il there belongs to every ele- ment V of the system li a determinate element ifi), for which ^ (r) = i// («'), we have gen- erally, III. il/{n')-^e considered in (132), (133), which changes every element n of jY into an element v of O, i. e., into ^{n). This element v can be called the «th element of O and accordingly the number n is itself the «th number of the number- 96 THE NA TURE AND series N. The same significance which the transfor- mation <^ possesses for the laws in the domain N, in so far as every element n is followed by a determinate element <^(«)^«', is found, after the change effected by 1^, to belong to the transformation 6 for the same laws in the domain O, in so far as the element v = i/f(«) arising from the change of n is followed by the ele- ment d{y)^^\l/{n') arising from the change of n'; we are therefore justified in saying that hy ij/ is changed into 0, which is symbolically expressed by d=^^t^^> m. Proof by complete induction (80). For p. by (135, II) and (91) the theorem is true for « = 1. 0-. If the theorem is true for a number n, then by (95) it is also true for the following number n' , be- cause by (135, III) and (91) m -\- ri ^={m -\- n)' ~^ m -\- n, which was to be proved. 143. Theorem. The conditions w > a and »« + «> a-\-n are equivalent. Proof by complete induction (80). For p. by (135, II) and (94) the theorem is true for n = \. u. If the theorem is true for a number «, then is it also true for the following number n, since by (94) the condition /« + « > a + ;z is equivalent to {m + «)'> (a+«)', hence by (135, III) also equivalent to »? + «'> a -|- «', which was to be proved. loo THE NATURE AND 144. Theorem. \im.'>a and n^b, then is also m -\- n';:> a -\- b . Proof. For from our hypotheses we have by (143) m^n^a^n and « + a > <5 + « or, what by (140) is the same, a + « > a + <5, whence the theorem follows by (95). 145. Theorem. If /« + « = « + «, then »?= a. Proof. For if m does not =a, hence by (90) either »2>a or m a+« or m-\-n«, then there exists one and by (157) only one number m which satisfies the con- dition m-\- n-z^l. Proof by complete induction (80). For p. the theorem is true for n^X. In fact, if />1, i. e., (89) if / is contained in N' , and hence is the transform m' of a number nt, then by (135, II) it fol- lows that l=m-\- 1, which was to be proved. 0-. If the theorem is true for a number n, then we show that it is also true for the following number «'. In fact, if /> «', then by (91), (95) also /> n, and hence there exists a number k which satisfies the condition l=^k-\- n ; since by (138) this is different from 1 (other- wise / would be = «' ) then by (78) is it the transform m' of a number m, consequently l^m' -\-n, therefore also by (136) l = m-\-n', which was to be proved. MEANING OF NUMBERS. loi XII. MULTIPLICATION OF NUMBERS. 147. Definition. After having found in XI an in- finite system of new transformations of the number- series iVin itself, we can by (126) use each of these in order to produce new transformations ^ of N. When we take fl^TV, and d{ny=m-\-n=n-\-m, where m.\s a determinate number, we certainly again have I. xf;{N)iN; and it remains, to determine i^ completely only to se- lect the element w from iV at pleasure. The simplest case occurs when we bring this choice into a certain agreement with the choice of 6, by putting w^m. Since the thus perfectly determinate i/^ depends upon this number m, we designate the corresponding trans- form ip{n) of any number « by the symbol my^n ox m.n or mn, and call this number the //-^^i/ac/ arising from the number m by multiplication by the number n, or in short the product of the numbers m, n. This therefore by (126) is completely determined by the conditions II. m.l=m III. mn' = mn-\- m, 148. Theorem, m' n^mnA^n. Proof by complete induction (80). For p. by (147, II) and (135, II) the theorem is, true for « = 1. 102 THE NA rURE AND a. If the theorem is true for a number n, we have m' n -\- tn' ^{mn -\- tC) -\- m' and consequently by (147, III), (141), (140), (136), (141), (147, III) m' n' = m n -\- (^n -\-fn') = mn-{- (m'-\- n)=^mn-\- (jn-\-n') = (mn-\- ni)-\- n' ^mn' -\- n'; therefore the theorem is true for the following num- ber «', which was to be proved. 149. Theorem, l.n^n. Proof by complete induction (80). For p. by (147, II) the theorem is true for n=^l. a-. If the theorem is true for a number n, then we have l.n-\-l--=n + l, i. e., by (147, III), (135, II) 1 .n' = n', therefore the theorem also holds for the fol- lowing number «', which was to be proved. 150. Theorem. mn = nm. Proof by complete induction (80). For p by (147, II), (149) the theorem is true for n = l. (T. If the theorem is true for a number n, then we have mn-\-m^nm-\-m, i. e., by (147, III), (148) mn' = n'm, therefore the the- orem is also true for the following number «', which was to be proved. 151. Theorem. l(m-\'n) = lm-\- In. Proof by complete induction (80). For p. by (135, II), (147, III), (147, II) the theorem is true for « = 1. MEANING OF NUMBERS. 103 0-. If the theorem is true for a number n, we have l{m -^ n)^ l={lm-\- ln)-\- 1; but by (147, III), (135, III) we have l{m -\-n)-\-l= l{m + n)' = l{m + n'), and by (141), (147, III) {Im + In) -{-l=lm-\- {In + l) = Im + In', consequently l{m-\- n') = lm-\- In', i. e., the theorem is true also for the following number n', which was to be proved. 152. Theorem. {m-\-n)l=zml-\-nl. The proof follows from (151), (150). 153. Theorem. (lni)n=^l(mn). Proof by complete induction (80). For p. by (147, II) the theorem is true for «z=l. (T. If the theorem is true for a number n, then we have (/w) n-\- lm = l{m n) -\- Im, i. e., by (147, III), (151), (147, III) {Jni)ri ^=l{mn-\- ni) :^ I (m n') , hence the theorem is also true for the following num- ber n', which was to be proved. 154. Remark. If in (147) we had assumed no re- lation between w and 6, but had put a = k, d(n) = m -\- n, then by (126) we should have had a less simple transformation i/r of the number-series N; for the num- ber 1 would \j/(l) = k and for every other number (therefore contained in the form «') would ^(«') = mn-\-k; since thus would be fulfilled, as one could I04 THE NATURE AND easily convince himself by the aid of the foregoing theorems, the condition y^in'^^Bx^^ri), i. e., \^(n")=. m-{-^{n) for all numbers n. XIII, INVOLUTION OF NUMBERS. 155. Definition. If in theorem (126) we again put Cl = N, and further »«, and therefore by (93), (74) belong to the chain m,; in fact, p. this is immediately evident for n = m, and (T. if this property belongs to a number n it follows again from III a,nd the nature of 6, that it also belongs to the number n', which was to be proved. After this special property of our new series of transformations }p„ has been established, we can easily prove our the- orem. 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