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You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.net Title: Essays on the Theory of Numbers Author: Richard Dedekind Translator: Wooster Woodruff Beman Release Date: April 8, 2007 [EBook #21016] Language: English Character set encoding: TeX *** START OF THE PROJECT GUTENBERG EBOOK THEORY OF NUMBERS *** Produced by Jonathan Ingram, Keith Edkins and the Online Distributed Proofreading Team at http://www.pgdp.net \end{verbatim} \normalsize \frontmatter \pagestyle {empty} \bigskip\bigskip \noindent \textsc{Transcriber's Note:} \emph{The symbol $\partof$ is used as an approximation to the author's Part-of symbol, not to be confused with the digit $3$. Internal page references have been been adjusted to fit the pagination of this edition. A few typographical errors have been corrected - these are noted at the very end of the text.} \newpage % [File: 001.png] \begin{center} {\large IN THE SAME SERIES.} \medskip \rule[.5ex]{2cm}{.2pt} \end{center} \smallskip \begin{small} \begin{itemize} \item[]\hspace{-2em}ON CONTINUITY AND IRRATIONAL NUMBERS, and ON THE NATURE AND MEANING OF NUMBERS\@. By \textsc{R.~Dedekind}. From the German by \textit{W.~W. Beman}. Pages, 115. Cloth, 75 cents net (3s.\ 6d.\ net). \item[]\hspace{-2em}GEOMETRIC EXERCISES IN PAPER-FOLDING\@. By \textsc{T. Sundara Row}. Edited and revised by \textit{W.~W. Beman} and \textit{D.~E. Smith}. With many half-tone engravings from photographs of actual exercises, and a package of papers for folding. Pages, circa 200. Cloth, \$1.00. net (4s.\ 6d.\ net). (In Preparation.) \item[]\hspace{-2em}ON THE STUDY AND DIFFICULTIES OF MATHEMATICS\@. By \textsc{Augustus De~Morgan}. Reprint edition with portrait and bibliographies. Pp., 288. Cloth, \$1.25 net (4s.\ 6d.\ net). \item[]\hspace{-2em}LECTURES ON ELEMENTARY MATHEMATICS\@. By \textsc{Joseph Louis Lagrange}. From the French by \textit{Thomas J. McCormack}. With portrait and biography. Pages, 172. Cloth, \$1.00 net (4s.\ 6d.\ net). \item[]\hspace{-2em}ELEMENTARY ILLUSTRATIONS OF THE DIFFERENTIAL AND INTEGRAL CALCULUS\@. By \textsc{Augustus De Morgan}. Reprint edition. With a bibliography of text-books of the Calculus. Pp., 144. Price, \$1.00 net (4s.\ 6d.\ net). \item[]\hspace{-2em}MATHEMATICAL ESSAYS AND RECREATIONS\@. By \textsc{Prof.\ Hermann Schubert}, of Hamburg, Germany. From the German by \textit{T.~J. McCormack}, Essays on Number, The Magic Square, The Fourth Dimension, The Squaring of the Circle. Pages, 149. Price, Cloth, 75c.\ net (3s.\ net). \item[]\hspace{-2em}A BRIEF HISTORY OF ELEMENTARY MATHEMATICS\@. By \textsc{Dr.\ Karl Fink}, of Tübingen. From the German by \textit{W.~W. Beman} and \textit{D.~E. Smith}, Pp.~333. Cloth, \$1.50 net (5s.\ 6d.\ net). \end{itemize} \end{small} \begin{center} \smallskip \rule[.5ex]{2cm}{.2pt} \medskip THE OPEN COURT PUBLISHING COMPANY {\scriptsize 324 DEARBORN ST., CHICAGO.} {\small LONDON: Kegan Paul, Trench, Trübner \& Co.} \end{center} % [File: 002.png] %title page \newpage \begin{center} {\huge ESSAYS} \vspace{0.1in} {\footnotesize ON THE} \vspace{0.1in} {\huge THEORY OF NUMBERS} \vspace{0.5in} I\@. CONTINUITY AND IRRATIONAL NUMBERS \smallskip II\@. THE NATURE AND MEANING OF NUMBERS \vspace{0.5in} {\footnotesize BY } \smallskip {\large RICHARD DEDEKIND} \vspace{0.5in} {\footnotesize AUTHORISED TRANSLATION BY} \smallskip WOOSTER WOODRUFF BEMAN \smallskip {\footnotesize PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF MICHIGAN} \vfill \rule[.5ex]{3cm}{1pt} \vfill {\large CHICAGO} THE OPEN COURT PUBLISHING COMPANY \smallskip {\footnotesize LONDON AGENTS} {\footnotesize \textsc{Kegan Paul, Trench, Trübner \& Co., Ltd.}} 1901 \end{center} % [File: 003.png] %copyright page \newpage \begin{center} TRANSLATION COPYRIGHTED \smallskip BY \smallskip \textsc{The Open Court Publishing Co.} 1901. \end{center} % [File: 004.png] %[**F1: ToC commented out, but proofed text left for comparison.] % %\tableofcontents % %CONTENTS. % %I. CONTINUITY AND IRRATIONAL NUMBERS. % %Page %Preface 1 % %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 19 % %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 101 % %XIII. Involution of Numbers 104 % %XIV. Number of the Elements of a Finite System 105 % \thispagestyle{empty} % [File: 005.png] \newpage \mainmatter \pagestyle{plain} %\part*{CONTINUITY AND IRRATIONAL NUMBERS} % [File: 006.png] %[Blank Page] % [File: 007.png] \mychap{I}{CONTINUITY AND IRRATIONAL NUMBERS} \addcontentsline{toc}{section}{Preface} My attention was first directed toward the considerations which form the subject of this pamphlet in the autumn of 1858. As professor in the Polytechnic School in Zürich 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 recourse 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 indispensable, if one does not wish to lose too much time. But that this form of introduction into the differential calculus 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 % [File: 008.png] purely arithmetic and perfectly rigorous foundation for the principles of infinitesimal analysis. The statement is so frequently made that the differential calculus deals with continuous magnitude, and yet an explanation of this continuity is nowhere given; even the most rigorous expositions of the differential calculus 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-mentioned theorem, and a more careful investigation convinced 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 Durège with whom I had a long and lively discussion. Later I explained these views of a scientific basis of arithmetic to a few of my pupils, and here in Braunschweig read a paper upon the subject before the scientific 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. Nevertheless % [File: 009.png---\Firefly Diva\\sahobart\glimpseofchaos\hutcheson\----- 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 \textit{Die Elemente der Funktionenlehre} by E.~Heine (\textit{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 indeed 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 \textit{Ueber die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen}, by G.~Cantor (\textit{Math. Annalen}, Vol.~5), for which I owe the ingenious author my hearty thanks. As I find on a hasty perusal, the axiom 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. \mysect{XVII}{I}{Properties of Rational Numbers} %\hspace{\parindent} The development of the arithmetic of rational numbers is here presupposed, but still I think it worth while to call attention to certain important % [File: 010.png---\Firefly Diva\\sahobart\glimpseofchaos\hutcheson\----- matters without discussion, so as to show at the outset the standpoint assumed in what follows. I regard the whole of arithmetic as a necessary, or at least natural, 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 consecutive new one to be formed. The chain of these numbers forms in itself an exceedingly useful instrument for the human mind; it presents an inexhaustible wealth of remarkable laws obtained by the introduction of the four fundamental operations of arithmetic. Addition is the combination of any arbitrary repetitions of the above-mentioned simplest act into a single act; from it in a similar way arises multiplication. While the performance of these two operations is always possible, that of the inverse operations, subtraction and division, proves to be limited. Whatever the immediate occasion may have been, whatever comparisons 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 creative 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 instrument of infinitely greater perfection. This system, % [File: 011.png---\Firefly Diva\\sahobart\glimpseofchaos\hutcheson\----- which I shall denote by $R$, possesses first of all a completeness and self-containedness which I have designated in another place\footnote{\textit{Vorlesungen über Zahlentheorie}, by P.~G.~Lejeune Dirichlet. 2d~ed. §159.} as characteristic of a \textit{body of numbers} [Zahlkörper] and which consists in this that the four fundamental operations are always performable 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 reason it will be necessary to bring out clearly the corresponding 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 $a = b$ as well as $b = a$. The fact that two rational numbers $a$, $b$ are 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 \textit{greater} than $b$, $b$ \textit{less} than $a$; this is also indicated by the symbols $a > b$, $b < a$.\footnote{Hence in what follows the so-called ``algebraic'' greater and less are understood unless the word ``absolute'' is added.} As in the latter case $b - a$ has a positive value it follows % [File: 012.png---\Firefly Diva\\sahobart\glimpseofchaos\hutcheson\----- that $b > a$, $a < b$. In regard to these two ways in which two numbers may differ the following laws will hold: \textsc{i.} If $a > b$, and $b > c$, 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 geometric ideas, express this briefly by saying: $b$ lies between the two numbers $a$, $c$. \textsc{ii.} If $a$, $c$ are two different numbers, there are infinitely many different numbers lying between $a$, $c$. \textsc{iii.} If $a$ is any definite number, then all numbers of the system $R$ fall into two classes, $A_1$ and $A_2$, each of which contains infinitely many individuals; the first class $A_1$ comprises all numbers $a_1$ that are $< a$, the second class $A_2$ comprises all numbers $a_2$ 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 $A_1$, $A_2$ is such that every number of the first class $A_1$ is less than every number of the second class $A_2$. %II. \mysect{XVI}{II}{Comparison of the Rational Numbers with the Points of a Straight Line} The above-mentioned properties of rational numbers recall the corresponding relations of position of % [File: 013.png] the points of a straight line $L$. If the two opposite directions existing upon it are distinguished by ``right'' and ``left,'' and $p$, $q$ are two different points, then either $p$ lies to the right of $q$, and at the same time $q$ to the left of $p$, or conversely $q$ lies to the right of $p$ and at the same time $p$ 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: \textsc{i.} If $p$ lies to the right of $q$, and $q$ to the right of $r$, then $p$ lies to the right of $r$; and we say that $q$ lies between the points $p$ and $r$. \textsc{ii.} If $p$, $r$ are two different points, then there always exist infinitely many points that lie between $p$ and $r$. \textsc{iii.} If $p$ is a definite point in $L$, then all points in $L$ fall into two classes, $P_1$, $P_2$, each of which contains infinitely many individuals; the first class $P_1$ contains all the points $p_1$, that lie to the left of $p$, and the second class $P_2$ contains all the points $p_2$ that lie to the right of $p$; the point $p$ itself may be assigned at pleasure to the first or second class. In every case the separation of the straight line $L$ into the two classes or portions $P_1$, $P_2$, is of such a character that every point of the first class $P_1$ lies to the left of every point of the second class $P_2$. 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 % [File: 014.png] 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 corresponding 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 $p$, 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 $p$, i.~e., an individual in $L$. To the two numbers $a$, $b$ respectively correspond the two points, $p$, $q$, and if $a>b$, then $p$ lies to the right of $q$. To the laws \textsc{i}, \textsc{ii}, \textsc{iii} of the previous Section correspond completely the laws \textsc{i}, \textsc{ii}, \textsc{iii} of the present. %III. \mysect{XV}{III}{Continuity of the Straight Line} \label{EISIII} 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 $p$ corresponds to the rational number $a$, then, as is well known, the length $o\,p$ is commensurable 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 % [File: 015.png] knew and had demonstrated that there are lengths incommensurable 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 incommensurable with the unit of length, we may affirm: The straight line $L$ is infinitely richer in point-individuals than the domain $R$ of rational numbers in number-individuals. If now, as is our desire, we try to follow up arithmetically all phenomena in the straight line, the domain 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 completeness, or as we may say at once, the same \textit{continuity}, as the straight line. The previous considerations are so familiar and well known to all that many will regard their repetition quite superfluous. Still I regarded this recapitulation as necessary to prepare properly for the main question. For, the way in which the irrational numbers 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 % [File: 016.png] of the same kind.\footnote{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 numbers.} Instead of this I demand that arithmetic shall be developed out of itself. That such comparisons with non-arithmetic notions have furnished the immediate occasion for the extension 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 endeavor 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 continuity. In what then does this continuity consist? Everything must depend on the answer to this question, and only through it shall we obtain a scientific basis for the investigation of \textit{all} continuous domains. By vague remarks upon the unbroken connection in % [File: 017.png] the smallest parts obviously nothing is gained; the problem is to indicate a precise characteristic of continuity 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 commonplace. It consists of the following. In the preceding section attention was called to the fact that every point $p$ of the straight line produces a separation 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 converse, 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 division 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 commonplace 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 % [File: 018.png] 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 existence it is \textit{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 accordance with the above principle. %IV. \mysect{XII}{IV}{Creation of Irrational Numbers} \label{EISIV} 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 $a_1$ of the first class $A_1$ is less than every number $a_2$ of the second class $A_2$; the number $a$ is either the greatest number of the class $A_1$ or the least number of the class $A_2$. If now any separation of the system $R$ into two classes $A_1$, $A_2$ is given which possesses only \textit{this} characteristic property that every number $a_1$ in $A_1$ is less than every number $a_2$ in $A_2$, then for brevity we shall call % [File: 019.png] such a separation a \textit{cut} [Schnitt] and designate it by $(A_1, A_2)$. We can then say that every rational number $a$ produces one cut or, strictly speaking, two cuts, which, however, we shall not look upon as essentially different; this cut possesses, \textit{besides}, the property that either among the numbers of the first class there exists a greatest or among the numbers of the second class a least number. And conversely, if a cut possesses this property, then it is produced by this greatest or least rational number. But it is easy to show that there exist infinitely many cuts not produced by rational numbers. The following example suggests itself most readily. Let $D$ be a positive integer but not the square of an integer, then there exists a positive integer $\lambda$ such that \[ \lambda^2 < D <(\lambda + 1)^2. \] If we assign to the second class $A_2$, every positive rational number $a_2$ whose square is $> D$, to the first class $A_1$ all other rational numbers $a_1$, this separation forms a cut $(A_1, A_2)$, i.~e., every number $a_1$ is less than every number $a_2$. For if $a_1 = 0$, or is negative, then on that ground $a_1$ is less than any number $a_2$, because, by definition, this last is positive; if $a_1$ is positive, then is its square $\leqq D$, and hence $a_1$ is less than any positive number $a_2$ whose square is $> D$. 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 $= D$. % [File: 020.png] 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 $= D$, then there exist two positive integers $t$, $u$, that satisfy the equation \[ t^2 - Du^2 = 0, \] and we may assume that $u$ is the \textit{least} positive integer possessing the property that its square, by multiplication by $D$, may be converted into the square of an integer $t$. Since evidently \[ \lambda u < t < (\lambda + 1) u, \] the number $u' = t - \lambda u$ is a positive integer certainly \textit{less} than $u$. If further we put \[ t' = Du - \lambda t, \] $t'$ is likewise a positive integer, and we have \[ t'^2 - Du'^2 = (\lambda^2 - D)(t^2 - Du^2) = 0, \] which is contrary to the assumption respecting $u$. Hence the square of every rational number $x$ is either $< D$ or $> D$. From this it easily follows that there is neither in the class $A_1$ a greatest, nor in the class $A_2$ a least number. For if we put \begin{align*} y &= \frac{x(x^2 + 3D)}{3x^2 + D}, \shortintertext {we have} y - x &= \frac{2x(D - x^2)}{3x^2 + D} \shortintertext {and} y^2 - D &= \frac{(x^2 - D)^3}{(3x^2 + D)^2}. \end{align*} % [File: 021.png] If in this we assume $x$ to be a positive number from the class $A_1$, then $x^2x$ and $y^2D$, and hence $y0$, and $y^2>D$. Therefore $y$ likewise belongs to the class $A_2$. 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 discontinuity of the domain $R$ of all rational numbers. Whenever, then, we have to do with a cut $(A_1, A_2)$ produced by no rational number, we create a new, an \textit{irrational} number $\alpha$, which we regard as completely defined by this cut $(A_1, A_2)$; we shall say that the number $\alpha$ corresponds to this cut, or that it produces this cut. From now on, therefore, to every definite cut there corresponds a definite rational or irrational number, and we regard two numbers as \textit{different} or \textit{unequal} always and only when they correspond to essentially different cuts. In order to obtain a basis for the orderly arrangement of all \textit{real}, i.~e., of all rational and irrational numbers we must investigate the relation between any two cuts $(A_1, A_2)$ and $(B_1, B_2)$ produced by any two numbers $\alpha$ and $\beta$. Obviously a cut $(A_1, A_2)$ is given completely when one of the two classes, e.~g., the first $A_1$ is known, because the second $A_2$ consists of all rational numbers not contained in $A_1$, and the characteristic property of such a first class lies in this % [File: 022.png] that if the number $a_1$ is contained in it, it also contains all numbers less than $a_1$. If now we compare two such first classes $A_1$, $B_1$ with each other, it may happen 1. That they are perfectly identical, i.~e., that every number contained in $A_1$ is also contained in $B_1$, and that every number contained in $B_1$ is also contained in $A_1$. In this case $A_2$ is necessarily identical with $B_2$, and the two cuts are perfectly identical, which we denote in symbols by $\alpha=\beta$ or $\beta=\alpha$. But if the two classes $A_1$, $B_1$ are not identical, then there exists in the one, e.~g., in $A_1$, a number $a'_1=b'_2$ not contained in the other $B_1$ and consequently found in $B_2$; hence all numbers $b_1$ contained in $B_1$ are certainly less than this number $a'_1=b'_2$ and therefore all numbers $b_1$ are contained in $A_1$. 2. If now this number $a'_1$ is the only one in $A_1$ that is not contained in $B_1$, then is every other number $a_1$ contained in $A_1$ also contained in $B_1$ and is consequently $ \beta$ as well as $\beta < \alpha$. It is to be noticed that this definition coincides completely with the one given earlier, when $\alpha$, $\beta$ are rational. The remaining possible cases are these: 4. If there exists in $B_1$ one and only one number $b'_1 = a'_2$, that is not contained in $A_1$ then the two cuts $(A_1, A_2)$ and $(B_1, B_2)$ are only unessentially different and they are produced by one and the same rational number $\alpha = a'_2 = b'_1 = \beta$. 5. But if there are in $B_1$ at least two numbers which are not contained in $A_1$, then $\beta > \alpha$, $\alpha < \beta$. 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 \textit{comparative} (greater, less) to designate % [File: 024.png] the relation between $\alpha$, $\beta$; 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 notions 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 $\alpha>\beta$ it is obvious that the less number $\beta$, if rational, certainly belongs to the class $A_1$; for since there is in $A_1$ a number $a'_1=b'_2$ which belongs to the class $B_2$, it follows that the number $\beta$, whether the greatest number in $B_1$ or the least in $B_2$ is certainly $\leqq a'_1$ and hence contained in $A_1$. Likewise it is obvious from $\alpha>\beta$ that the greater number $\alpha$, if rational, certainly belongs to the class $B_2$, because $\alpha \geqq a'_1$. Combining these two considerations we get the following result: If a cut is produced by the number $\alpha$ then any rational number belongs to the class $A_1$ or to the class $A_2$ according as it is less or greater than $\alpha$; if the number $\alpha$ is itself rational it may belong to either class. From this we obtain finally the following: If $\alpha>\beta$, i.~e., if there are infinitely many numbers in $A_1$ not contained in $B_1$ then there are infinitely many such numbers that at the same time are different from $\alpha$ and from $\beta$; every such rational number $c$ is $<\alpha$, because % [File: 025.png] it is contained in $A_1$ and at the same time it is $>\beta$ because contained in $B_2$. %V. \mysect{XIII}{V}{Continuity of the Domain of Real Numbers} In consequence of the distinctions just established the system $\mathfrak{R}$ of all real numbers forms a well-arranged domain of one dimension; this is to mean merely that the following laws prevail: \textsc{i.} If $\alpha>\beta$, and $\beta>\gamma$, then is also $\alpha>\gamma$. We shall say that the number $\beta$ lies between $\alpha$ and $\gamma$. \textsc{ii.} If $\alpha$, $\gamma$ are any two different numbers, then there exist infinitely many different numbers $\beta$ lying between $\alpha$, $\gamma$. \textsc{iii.} If $\alpha$ is any definite number then all numbers of the system $\mathfrak{R}$ fall into two classes $\mathfrak{A}_1$ and $\mathfrak{A}_2$ each of which contains infinitely many individuals; the first class $\mathfrak{A}_1$ comprises all the numbers $\alpha_1$ that are less than $\alpha$, the second $\mathfrak{A}_2$ comprises all the numbers $\alpha_2$ that are greater than $\alpha$; the number $\alpha$ 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 $\mathfrak{R}$ into the two classes $\mathfrak{A}_1$, $\mathfrak{A}_2$ is such that every number of first class $\mathfrak{A}_1$ is smaller than every number of the second class $\mathfrak{A}_2$ and we say that this separation is produced by the number $\alpha$. For brevity and in order not to weary the reader I suppress the proofs of these theorems which follow % [File: 026.png] immediately from the definitions of the previous section. Beside these properties, however, the domain $\mathfrak{R}$ possesses also \textit{continuity}; i.~e., the following theorem is true: \textsc{iv.} \label{EISViv}If the system $\mathfrak{R}$ of all real numbers breaks up into two classes $\mathfrak{A}_1$, $\mathfrak{A}_2$ such that every number $\alpha_1$ of the class $\mathfrak{A}_1$ is less than every number $\alpha_2$ of the class $\mathfrak{A}_2$ then there exists one and only one number $\alpha$ by which this separation is produced. \textit{Proof.} By the separation or the cut of $\mathfrak{R}$ into $\mathfrak{A}_1$ and $\mathfrak{A}_2$ we obtain at the same time a cut $(A_1, A_2)$ of the system $R$ of all rational numbers which is defined by this that $A_1$ contains all rational numbers of the class $\mathfrak{A}_1$ and $A_2$ all other rational numbers, i.~e., all rational numbers of the class $\mathfrak{A}_2$. Let $\alpha$ be the perfectly definite number which produces this cut $(A_1, A_2)$. If $\beta$ is any number different from $\alpha$, there are always infinitely many rational numbers $c$ lying between $\alpha$ and $\beta$. If $\beta<\alpha$, then $c<\alpha$; hence $c$ belongs to the class $A_1$ and consequently also to the class $\mathfrak{A}_1$, and since at the same time $\beta\alpha$, then is $c>\alpha$; hence $c$ belongs to the class $A_2$ and consequently also to the class $\mathfrak{A}_2$, and since at the same time $\beta>c$, then $\beta$ also belongs to the same class $\mathfrak{A}_2$, because every number in $\mathfrak{A}_1$ is less than every number $c$ in $\mathfrak{A}_2$. Hence every number $\beta$ different % [File: 027.png] from $\alpha$ belongs to the class $\mathfrak{A}_1$ or to the class $\mathfrak{A}_2$ according as $\beta<\alpha$ or $\beta>\alpha$; consequently $\alpha$ itself is either the greatest number in $\mathfrak{A}_1$ or the least number in $\mathfrak{A}_2$, i.~e., $\alpha$ is one and obviously the only number by which the separation of $R$ into the classes $\mathfrak{A}_1$, $\mathfrak{A}_2$ is produced. Which was to be proved. %VI. \mysect{XII}{VI}{Operations with Real Numbers} \label{EISVI} To reduce any operation with two real numbers $\alpha$, $\beta$ to operations with rational numbers, it is only necessary from the cuts $(A_1, A_2)$, $(B_1, B_2)$ produced by the numbers $\alpha$ and $\beta$ in the system $R$ to define the cut $(C_1, C_2)$ which is to correspond to the result of the operation, $\gamma$. I confine myself here to the discussion of the simplest case, that of addition. If $c$ is any rational number, we put it into the class $C_1$, provided there are two numbers one $a_1$ in $A_1$ and one $b_1$ in $B_1$ such that their sum $a_1+b_1 \geqq c$; all other rational numbers shall be put into the class $C_2$. This separation of all rational numbers into the two classes $C_1$, $C_2$ evidently forms a cut, since every number $c_1$ in $C_1$ is less than every number $c_2$ in $C_2$. If both $\alpha$ and $\beta$ are rational, then every number $c_1$ contained in $C_1$ is $\leqq\alpha+\beta$, because $a_1\leqq\alpha$, $b_1\leqq\beta$, and therefore $a_1+b_1 \leqq\alpha+\beta$; further, if there were contained in $C_2$ a number $c_2<\alpha+\beta$, hence $\alpha+\beta=c_2+p$, where $p$ is a positive rational number, then we should have \[ c_2=(\alpha- \frac{1}{2} p)+(\beta- \frac{1}{2} p), \] % [File: 028.png] which contradicts the definition of the number $c_2$, because $\alpha- \frac{1}{2} p$ is a number in $A_1$, and $\beta- \frac{1}{2} p$ a number in $B_1$; consequently every number $c_2$ contained in $C_2$ is $\geqq\alpha+\beta$. Therefore in this case the cut $(C_1, C_2)$ is produced by the sum $\alpha+\beta$. Thus we shall not violate the definition which holds in the arithmetic of rational numbers if in all cases we understand by the sum $\alpha+\beta$ of any two real numbers $\alpha$, $\beta$ that number $\gamma$ by which the cut $(C_1, C_2)$ is produced. Further, if only one of the two numbers $\alpha$, $\beta$ is rational, e.~g., $\alpha$, it is easy to see that it makes no difference with the sum $\gamma=\alpha+\beta$ whether the number $\alpha$ is put into the class $A_1$ or into the class $A_2$. Just as addition is defined, so can the other operations of the so-called elementary arithmetic be defined, viz., the formation of differences, products, quotients, powers, roots, logarithms, and in this way we arrive at real proofs of theorems (as, e.~g., $\sqrt{2}\cdot\sqrt{3} =\sqrt{6}$), 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 \textit{interval}, i.~e., a system $A$ of rational numbers possessing the following 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 % [File: 029.png] cut are intervals. If there exist a rational number $a_1$ which is less and a rational number $a_2$ which is greater than every number of the interval $A$, then $A$ is called a finite interval; there then exist infinitely many numbers in the same condition as $a_1$ and infinitely many in the same condition as $a_2$; the whole domain $R$ breaks up into three parts $A_1$, $A$, $A_2$ and there enter two perfectly definite rational or irrational numbers $\alpha_1$, $\alpha_2$ which may be called respectively the lower and upper (or the less and greater) \textit{limits} of the interval; the lower limit $\alpha_1$ is determined by the cut for which the system $A_1$ forms the first class and the upper $\alpha_2$ by the cut for which the system $A_2$ forms the second class. Of every rational or irrational number $\alpha$ lying between $\alpha_1$ and $\alpha_2$ it may be said that it lies \textit{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 theorem $(a+b)c=ac+bc$) 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 $\lambda$ is the result of an operation performed on the numbers $\alpha$, $\beta$, $\gamma$, $\ldots$ and $\lambda$ lies within the interval $L$, then intervals $A$, $B$, $C$, $\ldots$ can be % [File: 030.png] taken within which lie the numbers $\alpha$, $\beta$, $\gamma$, $\ldots$ such that the result of the same operation in which the numbers $\alpha$, $\beta$, $\gamma$, $\ldots$ are replaced by arbitrary numbers of the intervals $A$, $B$, $C$, $\ldots$ is always a number lying within the interval $L$.'' The forbidding clumsiness, 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 \textit{variable magnitudes}, \textit{functions}, \textit{limiting values}, and it would be best to base the definitions of even the simplest arithmetic operations upon these ideas, a matter which, however, cannot be carried further here. %Observed to be close to bottom of page so force skip \newpage %VII. \mysect{XI}{VII}{Infinitesimal Analysis} Here at the close we ought to explain the connection 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 approaches a fixed limiting value $\alpha$ when in the course of the process $x$ lies finally between two numbers between which $\alpha$ itself lies, or, what amounts to the same, when the difference $x-\alpha$ taken absolutely becomes 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 $x$ grows % [File: 031.png] 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 $\alpha_2$ such that $x$ remains continually $<\alpha_2$; I designate by $\mathfrak{A}_2$ the system of all these numbers $\alpha_2$, by $\mathfrak{A}_1$ the system of all other numbers $\alpha_1$; each of the latter possesses the property that in the course of the process $x$ becomes finally $\geqq \alpha_1$, hence every number $\alpha_1$ is less than every number $\alpha_2$ and consequently there exists a number $\alpha$ which is either the greatest in $\mathfrak{A}_1$ or the least in $\mathfrak{A}_2$ (V, \textsc{iv}). The former cannot be the case since $x$ never ceases to grow, hence $\alpha$ is the least number in $\mathfrak{A}_2$. Whatever number $\alpha_1$ be taken we shall have finally $\alpha_10$), then by hypothesis a time will come after which $x$ will change by less than $\delta$, i.~e., if at this time $x$ has the value $a$, then afterwards we shall continually have $x>a-\delta$ and $x\alpha_2$. Since every number $\alpha_1$ is less than every number $\alpha_2$ there exists a perfectly definite number $\alpha$ which produces this cut $(\mathfrak{A}_1, \mathfrak{A}_2)$ of the system $\mathfrak{R}$ 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 $(\mathfrak{B}_1, \mathfrak{B}_2)$ of the system $\mathfrak{R}$ is produced; a number $\beta_2$ (e.~g., $a-\delta$) is assigned to $\mathfrak{B}_2$ when in the course of the process $x$ becomes finally $\geqq\beta$; every other number $\beta_2$, to be assigned to $\mathfrak{B}_2$, has the property that $x$ is never % [File: 033.png] finally $\geqq \beta_2$; therefore infinitely many times $x$ becomes $<\beta_2$; the number $\beta$ by which this cut is produced I call the lower limiting value of the variable $x$. The two numbers $\alpha$, $\beta$ are obviously characterised by the following property: if $\epsilon$ is an arbitrarily small positive magnitude then we have always finally $x < \alpha + \epsilon$ and $x > \beta - \epsilon$, but never finally $x < \alpha - \epsilon$ and never finally $x > \beta + \epsilon$. Now two cases are possible. If $\alpha$ and $\beta$ are different from each other, then necessarily $\alpha > \beta$, since continually $\alpha_2 \geqq \beta_2$; the variable $x$ oscillates, and, however far the process advances, always undergoes changes whose amount surpasses the value $(\alpha - \beta) -2\epsilon$ where $\epsilon$ is an arbitrarily small positive magnitude. The original hypothesis to which I now return contradicts this consequence; there remains only the second case $\alpha = \beta$ since it has already been shown that, however small be the positive magnitude $\epsilon$, we always have finally $x < \alpha + \epsilon$ and $x > \beta - \epsilon$, $x$ approaches the limiting value $\alpha$, which was to be proved. These examples may suffice to bring out the connection between the principle of continuity and infinitesimal analysis. % [File: 034.png] %\part*{THE NATURE AND MEANING OF NUMBERS} % [File: 035.png] %[Blank Page] % [File: 036.png] \newpage \mychap{II}{THE NATURE AND MEANING OF NUMBERS} \addcontentsline{toc}{section}{Prefaces} \begin{center} {\large PREFACE TO THE FIRST EDITION.} \end{center} In science nothing capable of proof ought to be accepted 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.\footnote{Of the works which have come under my observation I mention the valuable \textit{Lehrbuch der Arithmetik und Algebra} of E.~Schröder (Leipzig, 1873), which contains a bibliography of the subject, and in addition the memoirs 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 independence of the works of others.} 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 intuitions of space and time, that I consider it an immediate 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 building up the science of numbers and by thus acquiring % [File: 037.png] 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.\footnote{See Section III. of my memoir, \textit{Continuity and Irrational Numbers} (Braunschweig, 1872), translated at pages \pageref{EISIII} et seq.\ of the present volume.} 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 correspond to a thing, or to represent a thing by a thing, an ability without which no thinking is possible. Upon this unique and therefore absolutely indispensable foundation, as I have already affirmed in an announcement of this paper,\footnote{Dirichlet's \textit{Vorlesungen über Zahlentheorie}, third edition, 1879, § 163, note on page 470.} 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 \textit{Continuity}, but only after its appearance and with many interruptions occasioned 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 mathematicians 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 [\textit{Anzahl}] of things (161), the % [File: 038.png] proof that the form of argument known as complete induction (or the inference from $n$ to $n + 1$) 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 possessing 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 faithful 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 dissection 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 evident and certain. On the contrary in just this possibility 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 performs % [File: 039.png] 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 therefore a very small effort or exertion of the mind, is sufficient for the practised reader to recognise the correct, true word, only with very great probability, to be sure; 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 increasing 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 continually 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 teachers 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 [\textit{Anzahl}] of things) are erroneously regarded as simple. In this sense which I wish to express by the word formed after a well-known saying \textit{\textgreek{\as e\ig{} \oR{} \asa njrwpos \as rijmht\ia zai}}, %$\alpha\epsilon\iota\;\omicron\;\alpha\nu\theta\rho\omega\pi\omicron\varsigma\;\alpha\rho\iota\theta\mu\eta\tau\iota\zeta\epsilon\iota$, I hope that the following pages, as an attempt to establish the science of numbers upon a uniform foundation will find a generous % [File: 040.png] 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 magnitudes, which according to my view can attain perfect clearness only through the science of numbers), this I have shown at least for irrational numbers in my former memoir on \textit{Continuity} (1872); in a way wholly similar, as I have already shown in Section~III. of that memoir,\footnote{Pages \pageref{EISIII} et seq.\ of the present volume.} 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 matter how remote, can be expressed as a theorem about natural numbers,---a declaration I have heard repeatedly 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 circumlocution and insisting on the use and recognition of no other than rational numbers. On the contrary, % [File: 041.png] 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 phenomena 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 Göttingen. 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 \textit{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.)\footnote{Pages \pageref{EISIV} et seq.\ of the present volume.} occurring in the domain of rational numbers which I designate by the term cut [\textit{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., \textsc{iv}.).\footnote{Page \pageref{EISViv} of the present volume.} 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 modification by U.~Dini in his \textit{Fondamenti per la teorica delle % [File: 042.png] funzioni di variabili 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 author discusses the existence of a measurable quantity corresponding to the cut, might easily lead to the supposition that my theory rests upon the consideration of such quantities. Nothing could be further from the truth; rather have I in Section III.\footnote{Pages \pageref{EISIII} et seq.\ of the present volume.} of my paper advanced several reasons why I wholly reject the introduction 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 mentioned by name but is never clearly defined and therefore cannot be employed in demonstrations. To explain 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,\footnote{Dirichlet's \textit{Vorlesungen über Zahlentheorie}, §~159 of the second edition, §~160 of the third.} 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 discontinuous; % [File: 043.png] but in spite of this discontinuity, and despite the existence of gaps in this space, all constructions that occur in Euclid's \textit{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 algebraic, 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 continuous space clear and definite. The same theory of irrational numbers founded upon the phenomenon of the cut is set forth in the \textit{Introduction à la théorie 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 \textit{Fondamenti} mentioned in the same preface, was unknown to him. % [File: 044.png] 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 \textit{Einleitung in die Differential- und Integralrechnung} (Leipzig, 1883). But I cannot quite agree with Tannery when he calls this theory the development of an idea due to J.~Bertrand and contained in his \textit{Traité d'arithmétique}, consisting in this that an irrational 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 \textit{Vorlesungen über allgemeine Arithmetik} (Leipzig, 1886), I venture to remark the following: That an irrational number is to be considered as fully defined by the specification just described, this conviction certainly long before the time of Bertrand 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 irrational 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 measurable 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 equality % [File: 045.png] of two ratios (\textit{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 mentioned above I wholly reject. Aside from this fact this method of presentation seems also in the succeeding 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.),\footnote {Pages \pageref{EISVI} et seq.\ of this volume.} that the theorem $\sqrt{2}\cdot\sqrt{3}=\sqrt{6}$ has nowhere 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. \begin{flushright} \textsc{R. Dedekind.}\mbox\qquad \end{flushright} \textsc{Harzburg}, October 5, 1887. % [File: 046.png] \newpage \begin{center} {\large PREFACE TO THE SECOND EDITION.} \end{center} The 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 definition of the infinite system had been pointed out before the appearance of my paper by G.~Cantor (\textit{Ein Beitrag zur Mannigfaltigkeitslehre}, \textit{Crelle's Journal}, Vol.~% 84, 1878), as also by Bolzano (\textit{Paradoxien des Unendlichen}, §~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 investigation, % [File: 047.png] 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 similarity 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 definition 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 nothing is said of all these things in either the one definition 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 \textit{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, points of very close contact with my paper, especially with my definition (44). The agreement, to be sure, is not easy to discover on account of the different form of expression; % [File: 048.png] but the positiveness with which the author speaks of the logical inference from $n$ to $n+1$ (page \pageref{Ind132}, below) shows plainly that here he stands upon the same ground with me. In the meantime E.~Schröder's \textit{Vorlesungen über 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. \begin{flushright} \textsc{R. Dedekind.}\mbox\qquad \end{flushright} \textsc{Harzburg}, August 24, 1893. % [File: 049.png] \newpage \begin{center} \large THE NATURE AND MEANING OF NUMBERS. \end{center} %I. \mysect{XVII}{I}{Systems of Elements} \mypara{1.} In what follows I understand by \textit{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 denoted 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 $b$), and $b$ the same as $a$, when all that can be thought concerning $a$ can also be thought concerning $b$, and when all that is true of $b$ can also be thought of $a$. That $a$ and $b$ are only symbols or names for one and the same thing is indicated by the notation $a=b$, and also by $b=a$. 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 different, $a$ is another thing than $b$, $b$ another thing than % [File: 050.png] $a$; there is some property belonging to the one that does not belong to the other. \mypara{2.} It very frequently happens that different things, $a$, $b$, $c$, \ldots\ 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 \textit{system} $S$; we call the things $a$, $b$, $c$, \ldots\ \textit{elements} of the system $S$, they are \textit{contained} in $S$; conversely, $S$ \textit{consists} of these elements. Such a system $S$ (an aggregate, a manifold, a totality) as an object of our thought is likewise 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.\footnote {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 Kronecker not long ago (\textit{Crelle's Journal}, Vol.~99, pp.~334--336) has endeavored to impose certain limitations 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 matter with more detail until the distinguished mathematician shall have published his reasons for the necessity or merely the expediency of these limitations.} The system $S$ 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 special case where a system $S$ consists of a \textit{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 element 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 % [File: 051.png] investigations it may be appropriate to imagine such a system. \mypara{3.} Definition. A system $A$ is said to be \textit{part} of a system $S$ when every element of $A$ is also element of $S$. Since this relation between a system $A$ and a system $S$ will occur continually in what follows, we shall express it briefly by the symbol $A \partof S$. The inverse symbol $S \wholeof A$, by which the same fact might be expressed, for simplicity and clearness I shall wholly avoid, but for lack of a better word I shall sometimes say that $S$ is \textit{whole} of $A$, by which I mean to express that among the elements of $S$ are found all the elements of $A$. Since further every element $s$ of a system $S$ by (2) can be itself regarded as a system, we can hereafter employ the notation $s \partof S$. \mypara{4.} Theorem. $A \partof A$, by reason of (3). \mypara{5.} Theorem. If $A \partof B$ and $B \partof A$, then $A = B$. The proof follows from (3), (2). \mypara{6.} Definition. A system $A$ is said to be a \textit{proper} [\textit{echter}] part of $S$, when $A$ is part of $S$, but different from $S$. According to (5) then $S$ is not a part of $A$, i.~e., there is in $S$ an element which is not an element of $A$. \mypara{7.} Theorem. If $A \partof B$ and $B \partof C$, which may be denoted briefly by $A \partof B \partof C$, then is $A \partof C$, and $A$ is certainly a proper part of $C$, if $A$ is a proper part of $B$ or if $B$ is a proper part of $C$. The proof follows from (3), (6). \mypara{8.} Definition. By the system \textit{compounded} out of % [File: 052.png] any systems $A$, $B$, $C$,~\ldots\ to be denoted by $\mathfrak{M}(A, B, C,\,\ldots)$ we mean that system whose elements are determined by the following prescription: a thing is considered as element of $\mathfrak{M}(A, B, C,\,\ldots)$ when and only when it is element of some one of the systems $A$, $B$, $C$,~\ldots, i.~e., when it is element of $A$, or $B$, or $C$,~\ldots. We include also the case where only a single system $A$ exists; then obviously $\mathfrak{M}(A)=A$. We observe further that the system $\mathfrak{M}(A, B, C,\,\ldots)$ compounded out of $A$, $B$, $C$,~\ldots\ is carefully to be distinguished from the system whose elements are the systems $A$, $B$, $C$,~\ldots\ themselves. \mypara{9.} Theorem. The systems $A$, $B$, $C$, \ldots\ are parts of $\mathfrak{M}(A, B, C,\,\ldots)$. The proof follows from (8), (3). \mypara{10.} Theorem. If $A$, $B$, $C$,~\ldots\ are parts of a system $S$, then is $\mathfrak{M}(A, B, C,\,\ldots)\\ \partof S$. The proof follows from (8), (3). \mypara{11.} Theorem. If $P$ is part of one of the systems $A$, $B$, $C$,~\ldots\ then is $P \partof \mathfrak{M}(A, B, C,\,\ldots)$. The proof follows from (9), (7). \mypara{12.} Theorem. If each of the systems $P$, $Q$,~\ldots\ is part of one of the systems $A$, $B$, $C$,~\ldots\ then is $\mathfrak{M}(P, Q,\,\ldots) \partof \mathfrak{M}(A, B, C,\,\ldots)$. The proof follows from (11), (10). \mypara{13.} Theorem. If $A$ is compounded out of any of the systems $P$, $Q$,~\ldots\ then is $A \partof \mathfrak{M}(P, Q,\,\ldots)$. Proof. For every element of $A$ is by (8) element of one of the systems $P$, $Q$,~\ldots, consequently by (8) % [File: 053.png] also element of $\mathfrak{M}(P, Q,\,\ldots)$, whence the theorem follows by (3). \mypara{14.} Theorem. If each of the systems $A$, $B$, $C$,~\ldots\ is compounded out of any of the systems $P$, $Q$,~\ldots\ then is \[ \mathfrak{M}(A,B,C,\,\ldots) \partof \mathfrak{M}(P,Q,\,\ldots) \] The proof follows from (13), (10). \mypara{15.} Theorem. If each of the systems $P$, $Q$,~\ldots\ is part of one of the systems $A$, $B$, $C$,~\ldots, and if each of the latter is compounded out of any of the former, then is \[ \mathfrak{M}(P, Q,\, \ldots) = \mathfrak{M}(A, B, C,\, \ldots). \] The proof follows from (12), (14), (5). \mypara{16.} Theorem. If \begin{gather*} A = \mathfrak{M}(P, Q) \mathrm{\ and\ } B = \mathfrak{M}(Q, R) \\ \mathrm{then\ is\ } \mathfrak{M}(A, R) = \mathfrak{M}(P, B). \end{gather*} Proof. For by the preceding theorem (15) \[ \mathfrak{M}(A, R) \mathrm{\ as\ well\ as\ } \mathfrak{M}(P, B) = \mathfrak{M}(P, Q, R). \] \mypara{17.} Definition. A thing $g$ is said to be \textit{common} element of the systems $A$, $B$, $C$,~\ldots, if it is contained in each of these systems (that is in $A$ \textit{and} in $B$ \textit{and} in $C$~\ldots). Likewise a system $T$ is said to be a \textit{common part} of $A$, $B$, $C$,~\ldots\ when $T$ is part of each of these systems; and by the \textit{community} [\textit{Gemeinheit}] of the systems $A$, $B$, $C$,~\ldots\ we understand the perfectly determinate system $\mathfrak{G}(A, B, C,\,\ldots)$ which consists of all the common elements $g$ of $A$, $B$, $C$,~\ldots\ and % [File: 054.png] hence is likewise a common part of those systems. We again include the case where only a single system $A$ occurs; then $\mathfrak{G}(A)$ (is to be put) $=A$. But the case may also occur that the systems $A$, $B$, $C$,~\ldots\ possess no common element at all, therefore no common part, no community; they are then called systems \textit{without} common part, and the symbol $\mathfrak{G}(A, B, C,\,\ldots)$ is meaningless (compare the end of (2)). We shall however almost always in theorems concerning communities leave it to the reader to add in thought the condition of their existence and to discover the proper interpretation of these theorems for the case of non-existence. \mypara{18.} Theorem. Every common part of $A$, $B$, $C$,~\ldots\ is part of $\mathfrak{G}(A, B, C,\,\ldots)$. The proof follows from (17). \mypara{19.} Theorem. Every part of $\mathfrak{G}(A, B, C,\,\ldots)$ is common part of $A$, $B$, $C$,~\ldots. The proof follows from (17), (7). \mypara{20.} Theorem. If each of the systems $A$, $B$, $C$,~\ldots\ is whole (3) of one of the systems $P$, $Q$,~\ldots\ then is \[ \mathfrak{G}(P, Q,\,\ldots) \partof \mathfrak{G}(A, B, C,\,\ldots) \] Proof. For every element of $\mathfrak{G}(P, Q,\,\ldots)$ is common element of $P$, $Q$,~\ldots, therefore also common element of $A$, $B$, $C$,~\ldots, which was to be proved. % [File: 055.png] %II. \mysect{XVI}{II}{Transformation of a System} \mypara{21.} Definition.\footnote {See Dirichlet's \textit{Vorlesungen über Zahlentheorie}, 3d edition, 1879, §~163.} By a \textit{transformation} [\textit{Abbildung}] $\phi$ of a system $S$ we understand a law according to which to every determinate element $s$ of $S$ there \textit{belongs} a determinate thing which is called the \textit{transform} of $s$ and denoted by $\phi(s)$; we say also that $\phi(s)$ \textit{corresponds} to the element $s$, that $\phi(s)$ \textit{results} or is \textit{produced} from $s$ by the transformation $\phi$, that $s$ is \textit{transformed} into $\phi(s)$ by the transformation $\phi$. If now $T$ is any part of $S$, then in the transformation $\phi$ of $S$ is likewise contained a determinate transformation of $T$, which for the sake of simplicity may be denoted by the same symbol $\phi$ and consists in this that to every element $t$ of the system $T$ there corresponds the same transform $\phi(t)$, which $t$ possesses as element of $S$; at the same time the system consisting of all transforms $\phi(t)$ shall be called the transform of $T$ and be denoted by $\phi(T)$, by which also the significance of $\phi(S)$ is defined. As an example of a transformation of a system we may regard the mere assignment of determinate symbols or names to its elements. The simplest transformation of a system is that by which each of its elements is transformed into itself; it will be called the \textit{identical} transformation of the system. For convenience, in the following theorems (22), (23), (24), which deal with an arbitrary transformation $\phi$ of % [File: 056.png] an arbitrary system $S$, we shall denote the transforms of elements $s$ and parts $T$ respectively by $s'$ and $T'$; in addition we agree that small and capital italics without accent shall always signify elements and parts of this system $S$. \mypara{22.} Theorem.\footnote{See theorem~27.} If $A \partof B$ then $A' \partof B'$. 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. \mypara{23.} Theorem. The transform of $\mathfrak{M}(A, B, C,\,\ldots)$ is $\mathfrak{M}(A', B', C',\,\ldots)$. Proof. If we denote the system $\mathfrak{M}(A, B, C,\, \ldots)$ which by (10) is likewise part of $S$ by $M$, then is every element of its transform $M'$ the transform $m'$ of an element $m$ of $M$; since therefore by (8) $m$ is also element of one of the systems $A$, $B$, $C$,~\ldots\ and consequently $m'$ element of one of the systems $A'$, $B'$, $C'$,~\ldots, and hence by (8) also element of $\mathfrak{M}(A', B', C',\, \ldots)$, we have by (3) \[ M' \partof \mathfrak{M}(A', B', C',\,\ldots). \] On the other hand, since $A$, $B$, $C$,~\ldots\ are by (9) parts of $M$, and hence $A'$, $B'$, $C'$,~\ldots\ by (22) parts of $M'$, we have by (10) \[ \mathfrak{M}(A', B', C',\,\ldots) \partof M'. \] By combination with the above we have by (5) the theorem to be proved \[ M' = \mathfrak{M}(A', B', C',\, \ldots). \] % [File: 057.png] \mypara{24.} Theorem.\footnote{See theorem~29.} The transform of every common part of $A$, $B$, $C$,~\ldots, and therefore that of the community $\mathfrak{G}(A, B, C,\,\ldots)$ is part of $\mathfrak{G}(A', B', C',\,\ldots)$. Proof. For by (22) it is common part of $A'$, $B'$, $C'$,~\ldots, whence the theorem follows by (18). \mypara{25.} Definition and theorem. If $\phi$ is a transformation of a system $S$, and $\psi$ a transformation of the transform $S'=\phi(S)$, there always results a transformation $\theta$ of S, \textit{compounded}\footnote {A confusion of this compounding of transformations with that of systems of elements is hardly to be feared.} out of $\phi$ and $\psi$, which consists of this that to every element $s$ of $S$ there corresponds the transform \[ \theta(s)=\psi(s')=\psi\bigl(\phi(s)\bigr), \] where again we have put $\phi(s)=s'$. This transformation $\theta$ can be denoted briefly by the symbol $\psi\centerdot\phi$ or $\psi\phi$, the transform $\theta(s)$ by $\psi\phi(s)$ where the order of the symbols $\phi$, $\psi$ is to be considered, since in general the symbol $\phi\psi$ has no interpretation and actually has meaning only when $\psi(s') \partof s$. If now $\chi$ signifies a transformation of the system $\psi(s')=\psi\phi(s)$ and $\eta$ the transformation $\chi\psi$ of the system $S'$ compounded out of $\psi$ and $\chi$, then is $\chi\theta(s)=\chi\psi(s')=\eta(s')=\eta\phi(s)$; therefore the compound transformations $\chi\theta$ and $\eta\phi$ coincide for every element $s$ of $S$, i.~e., $\chi\theta = \eta \phi$. In accordance with the meaning of $\theta$ and $\eta$ this theorem can finally be expressed in the form \[ \chi\centerdot\psi\phi = \chi\psi\centerdot\phi, \] % [File: 058.png] and this transformation compounded out of $\phi$, $\psi$, $\chi$ can be denoted briefly by $\chi\psi\phi$. %III. \mysect{XV}{III}{Similarity of a Transformation. Similar Systems} \mypara{26.} Definition. A transformation $\phi$ of a system $S$ is said to be \textit{similar} [\textit{ähnlich}] or \textit{distinct}, when to different elements $a$, $b$ of the system $S$ there always correspond different transforms $a' = \phi(a)$, $b' = \phi(b)$. Since in this case conversely from $s' = t'$ we always have $s = t$, then is every element of the system $S' = \phi(S)$ the transform $s'$ of a single, perfectly determinate element $s$ of the system $S$, and we can therefore set over against the transformation $\phi$ of $S$ an \textit{inverse} transformation of the system $S'$, to be denoted by $\overline{\phi}$, which consists in this that to every element $s'$ of $S'$ there corresponds the transform $\overline{\phi}(s') = s$, and obviously this transformation is also similar. It is clear that $\overline{\phi}(S') = S$, that further $\phi$ is the inverse transformation belonging to $\overline{\phi}$ and that the transformation $\overline{\phi}\phi$ compounded out of $\phi$ and $\overline{\phi}$ by (25) is the identical transformation of $S$ (21). At once we have the following additions to II., retaining the notation there given. \mypara{27.} Theorem.\footnote{See theorem~22.} If $A' \partof B'$, then $A \partof B$. Proof. For if $a$ is an element of $A$ then is $a'$ an element of $A'$, therefore also of $B'$, hence $=b'$, where $b$ is an element of $B$; but since from $a' = b'$ we always % [File: 059.png] have $a = b$, then is every element of $A$ also element of $B$, which was to be proved. \mypara{28.} Theorem. If $A' = B'$, then $A = B$. The proof follows from (27), (4), (5). \mypara{29.} Theorem.\footnote{See theorem~24.} If $G = \mathfrak{G}(A, B, C,\, \ldots)$, then \[ G' = \mathfrak{G}(A', B', C',\, \ldots). \] Proof. Every element of $\mathfrak{G}(A', B', C',\, \ldots)$ is certainly contained in $S'$, and is therefore the transform $g'$ of an element $g$ contained in $S$; but since $g'$ is common element of $A'$, $B'$, $C'$,~\ldots\ then by (27) must $g$ be common element of $A$, $B$, $C$,~\ldots\ therefore also element of $G$; hence every element of $\mathfrak{G}(A', B', C',\, \ldots)$ is transform of an element $g$ of $G$, therefore element of $G'$, i.~e., $\mathfrak{G}(A', B', C',\,\ldots) \partof G'$, and accordingly our theorem follows from (24), (5). \mypara{30.} Theorem. The identical transformation of a system is always a similar transformation. \mypara{31.} Theorem. If $\phi$ is a similar transformation of $S$ and $\psi$ a similar transformation of $\phi(S)$, then is the transformation $\psi\phi$ of $S$, compounded of $\phi$ and $\psi$, a similar transformation, and the associated inverse transformation $\overline{\psi}\,\overline{\phi} = \overline{\phi}\,\overline{\psi}$. Proof. For to different elements $a$, $b$ of $S$ correspond different transforms $a' = \phi(a)$, $b'=\phi(b)$, and to these again different transforms $\psi(a')=\psi\phi(a)$, $\psi(b') =\psi\phi(b)$ and therefore $\psi\phi$ is a similar transformation. Besides every element $\psi\phi(s) = \psi(s')$ of the system $\psi\phi(S)$ is transformed by $\overline{\psi}$ into $s' = \phi(s)$ and % [File: 060.png] this by $\overline{\phi}$ into $s$, therefore $\psi\phi(s)$ is transformed by $\overline{\phi}\,\overline{\psi}$ into $s$, which was to be proved. \mypara{32.} Definition. The systems $R$, $S$ are said to be \textit{similar} when there exists such a similar transformation $\phi$ of $S$ that $\phi(S)=R$, and therefore $\overline{\phi}(R) = S$. Obviously by (30) every system is similar to itself. \mypara{33.} Theorem. If $R$, $S$ are similar systems, then every system $Q$ similar to $R$ is also similar to $S$. Proof. For if $\phi$, $\psi$ are similar transformations of $S$, $R$ such that $\phi(S)=R$, $\psi(R)=Q$, then by (31) $\psi\phi$ is a similar transformation of $S$ such that $\psi\phi(S) = Q$, which was to be proved. \mypara{34.} Definition. We can therefore separate all systems into \textit{classes} by putting into a determinate class all systems $Q$, $R$, $S$,~\ldots, and only those, that are similar to a determinate system $R$, the \textit{representative} of the class; according to (33) the class is not changed by taking as representative any other system belonging to it. \mypara{35.} Theorem. If $R$, $S$ are similar systems, then is every part of $S$ also similar to a part of $R$, every proper part of $S$ also similar to a proper part of $R$. Proof. For if $\phi$ is a similar transformation of $S$, $\phi(S)=R$, and $T \partof S$, then by (22) is the system similar to $T$ $\phi(T) \partof R$; if further $T$ is proper part of $S$, and $s$ an element of $S$ not contained in $T$, then by (27) the element $\phi(s)$ contained in $R$ cannot be contained in $\phi(T)$; hence $\phi(T)$ is proper part of $R$, which was to be proved. % [File: 061.png] %IV. \mysect{XII}{IV}{Transformation of a System in Itself} \mypara{36.} Definition. If $\phi$ is a similar or dissimilar transformation of a system $S$, and $\phi(S)$ part of a system $Z$, then $\phi$ is said to be a transformation of $S$ \textit{in} $Z$, and we say $S$ is transformed by $\phi$ in $Z$. Hence we call $\phi$ a transformation of the system $S$ \textit{in itself}, when $\phi(S) \partof S$, and we propose in this paragraph to investigate the general laws of such a transformation $\phi$. In doing this we shall use the same notations as in II. and again put $\phi(s)=s'$, $\phi(T)=T'$. These transforms $s'$, $T'$ are by (22), (7) themselves again elements or parts of $S$, like all things designated by italic letters. \mypara{37.} Definition. $K$ is called a \textit{chain} [\textit{Kette}], when $K' \partof K$. 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 transformation $\phi$; with reference to another transformation of the system $S$ in itself $K$ can very well not be a chain. \mypara{38.} Theorem. $S$ is a chain. \mypara{39.} Theorem. The transform $K'$ of a chain $K$ is a chain. Proof. For from $K' \partof K$ it follows by (22) that $(K')' \partof K'$, which was to be proved. \mypara{40.} Theorem. If $A$ is part of a chain $K$, then is also $A' \partof K$. % [File: 062.png] Proof. For from $A \partof K$ it follows by (22) that $A' \partof K'$, and since by (37) $K' \partof K$, therefore by (7) $A' \partof K$, which was to be proved. \mypara{41.} Theorem. If the transform $A'$ is part of a chain $L$, then is there a chain $K$, which satisfies the conditions $A \partof K$, $K' \partof L$; and $\mathfrak{M}(A,L)$ is just such a chain $K$. Proof. If we actually put $K=\mathfrak{M}(A, L)$, then by (9) the one condition $A \partof K$ is fulfilled. Since further by (23) $K' = \mathfrak{M}(A',L')$ and by hypothesis $A' \partof L$, $L' \partof L$, then by (10) is the other condition $K' \partof L$ also fulfilled and hence it follows because by (9) $L \partof K$, that also $K' \partof K$, i.~e., $K$ is a chain, which was to be proved. \mypara{42.} Theorem. A system $M$ compounded simply out of chains $A$, $B$, $C$,~\ldots\ is a chain. Proof. \label{chain}Since by (23) $M'= \mathfrak{M}(A', B', C',\, \ldots)$ and by hypothesis $A' \partof A$, %[**Typo corrected, see note at end.] $B' \partof B$, $C' \partof C$,~\ldots\ therefore by (12) $M' \partof M$, which was to be proved. \mypara{43.} Theorem. The community $G$ of chains $A$, %[**Comma added, assumed just poor image.] $B$, $C$,~\ldots\ is a chain. Proof. Since by (17) $G$ is common part of $A$, $B$, $C$,~\ldots, therefore by (22) $G'$ common part of $A'$, $B'$, $C'$,~\ldots, and by hypothesis $A' \partof A$, $B' \partof B$, $C' \partof C$,~\ldots, then by (7) $G'$ is also common part of $A$, $B$, $C$,~\ldots\ and hence by (18) also part of $G$, which was to be proved. \mypara{44.} Definition. If $A$ is any part of $S$, we will denote by $A_0$ the community of all those chains (e.~g., $S$) % [File: 063.png] of which $A$ is part; this community $A_0$ exists (17) because $A$ is itself common part of all these chains. Since further by (43) $A_0$ is a chain, we will call $A_0$ the \textit{chain of the system} $A$, or briefly the chain of $A$. This definition too is strictly related to the fundamental determinate transformation $\phi$ of the system $S$ in itself, and if later, for the sake of clearness, it is necessary we shall at pleasure use the symbol $\phi_0(A)$ instead of $A_0$, and likewise designate the chain of $A$ corresponding to another transformation $\omega$ by $\omega_0(A)$. For this very important notion the following theorems hold true. \mypara{45.} Theorem. $A \partof A_0$. Proof. For $A$ is common part of all those chains whose community is $A_0$, whence the theorem follows by (18). \mypara{46.} Theorem. $(A_0)' \partof A_0$. Proof. For by (44) $A_0$ is a chain (37). \mypara{47.} Theorem. If $A$ is part of a chain $K$, then is also $A_0 \partof K$. Proof. For $A_0$ is the community and hence also a common part of all the chains $K$, of which $A$ is part. \mypara{48.} Remark. One can easily convince himself that the notion of the chain $A_0$ defined in (44) is completely characterised by the preceding theorems, (45), (46), (47). \mypara{49.} Theorem. $A' \partof (A_0)'$. The proof follows from (45), (22). % [File: 064.png] \mypara{50.} Theorem. $A' \partof A_0$. The proof follows from (49), (46), (7). \mypara{51.} Theorem. If $A$ is a chain, then $A_0=A$. Proof. Since $A$ is part of the chain $A$, then by (47) $A_0 \partof A$, whence the theorem follows by (45), (5). \mypara{52.} Theorem. If $B \partof A$, then $B \partof A_0$. The proof follows from (45), (7). \mypara{53.} Theorem. If $B \partof A_0$, then $B_0 \partof A_0$, and conversely. Proof. Because $A_0$ is a chain, then by (47) from $B \partof A_0$, we also get $B_0 \partof A_0$; conversely, if $B_0 \partof A_0$, then by (7) we also get $B \partof A_0$, because by (45) $B \partof B_0$. \mypara{54.} Theorem. If $B \partof A$, then is $B_0 \partof A_0$. The proof follows from (52), (53). \mypara{55.} Theorem. If $B \partof A_0$, then is also $B' \partof A_0$. Proof. For by (53) $B_0 \partof A_0$, and since by (50) $B' \partof B_0$, 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). \mypara{56.} Theorem. If $B \partof A_0$, then is $(B_0)' \partof (A_0)'$. The proof follows from (53), (22). \mypara{57.} Theorem and definition. $(A_0)'=(A')_0$, i.~e., the transform of the chain of $A$ is at the same time the chain of the transform of $A$. Hence we can designate this system in short by $A'_0$ and at pleasure call it the \textit{chain-transform} or \textit{transform-chain} of $A$. With the clearer notation given in (44) the theorem might be expressed by $\phi\bigl(\phi_0(A)\bigr) = \phi_0\bigl(\phi(A)\bigr)$. Proof. If for brevity we put $(A')_0=L$, $L$ is a % [File: 065.png] chain (44) and by (45) $A' \partof L$; hence by (41) there exists a chain $K$ satisfying the conditions $A \partof K$, $K' \partof L$; hence from (47) we have $A_0 \partof K$, therefore $(A_0)' \partof K'$, and hence by (7) also $(A_0)' \partof L$, i.~e., \[ (A_0)' \partof (A')_0. \] Since further by (49) $A' \partof (A_0)'$, and by (44), (39) $(A_0)'$ is a chain, then by (47) also \[ (A')_0 \partof (A_0)', \] whence the theorem follows by combining with the preceding result (5). \mypara{58.} Theorem. $A_0=\mathfrak{M}(A, A'_0)$, i.~e., the chain of $A$ is compounded out of $A$ and the transform-chain of $A$. Proof. If for brevity we again put \[ L=A'_0 =(A_0)'=(A')_0 \text{\ and\ } K=\mathfrak{M}(A, L), \] then by (45) $A' \partof L$, and since $L$ is a chain, by (41) the same thing is true of $K$; since further $A \partof K$ (9), therefore by (47) \[ A_0 \partof K. \] On the other hand, since by (45) $A \partof A_0$, and by (46) also $L \partof A_0$, then by (10) also \[ K \partof A_0, \] whence the theorem to be proved $A_0=K$ follows by combining with the preceding result (5). \mypara{59.} Theorem of complete induction. In order to show that the chain $A_0$ is part of any system $\Sigma$---be this latter part of $S$ or not---it is sufficient to show, $\rho$. that $A \partof \Sigma$, and % [File: 066.png] $\sigma$. that the transform of every common element of $A_0$ and $\Sigma$ is likewise element of $\Sigma$. Proof. For if $\rho$ is true, then by (45) the community $G=\mathfrak{G}(A_0,\Sigma)$ certainly exists, and by (18) $A \partof G$; since besides by (17) \[ G \partof A_0, \] then is $G$ also part of our system $S$, which by $\phi$ is transformed in itself and at once by (55) we have also $G' \partof A_0$. If then $\sigma$ is likewise true, i.~e.\ if $G' \partof \Sigma$, then must $G'$ as common part of the systems $A_0$, $\Sigma$ by (18) be part of their community $G$, i.~e. $G$ is a chain (37), and since, as above noted, $A \partof G$, then by (47) is also \[ A_0 \partof G, \] and therefore by combination with the preceding result $G=A_0$, hence by (17) also $A_0 \partof \Sigma$, which was to be proved. \mypara{60.} The preceding theorem, as will be shown later, forms the scientific basis for the form of demonstration known by the name of complete induction (the inference from $n$ to $n+1$); it can also be stated in the following manner: In order to show that all elements of the chain $A_0$ possess a certain property $\mathfrak{E}$ (or that a theorem $\mathfrak{S}$ dealing with an undetermined thing $n$ actually holds good for all elements $n$ of the chain $A_0$) it is sufficient to show $\rho$. that all elements $a$ of the system $A$ possess the property $\mathfrak{E}$ (or that $\mathfrak{S}$ holds for all $a$'s) and $\sigma$. that to the transform $n'$ of every such element $n$ of $A_0$ possessing the property $\mathfrak{E}$, belongs the same % [File: 067.png] property $\mathfrak{E}$ (or that the theorem $\mathfrak{S}$, as soon as it holds for an element $n$ of $A_0$, certainly must also hold for its transform $n'$). Indeed, if we denote by $\Sigma$ the system of all things possessing the property $\mathfrak{E}$ (or for which the theorem $\mathfrak{S}$ holds) the complete agreement of the present manner of stating the theorem with that employed in (59) is immediately obvious. \mypara{61.} Theorem. The chain of $\mathfrak{M}(A, B, C,\, \ldots)$ is $\mathfrak{M}(A_0,B_0,C_0,\,\ldots)$. Proof. If we designate by $M$ the former, by $K$ the latter system, then by (42) $K$ is a chain. Since then by (45) each of the systems $A$, $B$, $C$,~\ldots\ is part of one of the systems $A_0$, $B_0$, $C_0$,~\ldots, and therefore by (12) $M \partof K$, then by (47) we also have \[ M_0 \partof K. \] On the other hand, since by (9) each of the systems $A$, $B$, $C$,~\ldots\ is part of $M$, and hence by (45), (7) also part of the chain $M_0$, then by (47) must also each of the systems $A_0$, $B_0$, $C_0$,~\ldots\ be part of $M_0$, therefore by (10) \[ K \partof M_0 \] whence by combination with the preceding result follows the theorem to be proved $M_0=K$ (5). \mypara{62.} Theorem. The chain of $\mathfrak{G}(A, B, C,\,\ldots)$ is part of $\mathfrak{G}(A_0, B_0, C_0,\,\ldots)$. Proof. If we designate by $G$ the former, by $K$ the latter system, then by (43) $K$ is a chain. Since then each of the systems $A_0$, $B_0$, $C_0$,~\ldots\ by (45) is whole % [File: 068.png] of one of the systems $A$, $B$, $C$,~\ldots, and hence by (20) $G \partof K$, therefore by (47) we obtain the theorem to be proved $G_0 \partof K$. \mypara{63.} Theorem. If $K' \partof L \partof K$, and therefore $K$ is a chain, $L$ is also a chain. If the same is proper part of $K$, and $U$ the system of all those elements of $K$ which are not contained in $L$, and if further the chain $U_0$ is proper part of $K$, and $V$ the system of all those elements of $K$ which are not contained in $U_0$, then is $K=\mathfrak{M}(U_0, V)$ and $L = \mathfrak{M}(U'_0, V)$. If finally $L=K'$ then $V \partof V'$. The proof of this theorem of which (as of the two preceding) we shall make no use may be left for the reader. %V. \mysect{XIII}{V}{The Finite and Infinite} \mypara{64.} Definition.\footnote {If one does not care to employ the notion of similar systems (32) he must say: $S$ is said to be infinite, when there is a proper part of $S$ (6) in which $S$ can be distinctly (similarly) transformed (26), (36). In this form I submitted the definition of the infinite which forms the core of my whole investigation in September, 1882, to G.~Cantor and several years earlier to Schwarz and Weber. All other attempts that have come to my knowledge to distinguish the infinite from the finite seem to me to have met with so little success that I think I may be permitted to forego any criticism of them.} A system $S$ is said to be \textit{infinite} when it is similar to a proper part of itself (32); in the contrary case $S$ is said to be a \textit{finite} system. \mypara{65.} Theorem. Every system consisting of a single element is finite. Proof. For such a system possesses no proper part (2), (6). % [File: 069.png---\michael.j.white\\sahobart\glimpseofchaos\wwoods\----- \mypara{66.} Theorem. There exist infinite systems. Proof.\footnote {A similar consideration is found in §~13 of the \textit{Paradoxien des Unendlichen} by Bolzano (Leipzig, 1851).} My own realm of thoughts, i.~e., the totality $S$ of all things, which can be objects of my thought, is infinite. For if $s$ signifies an element of $S$, then is the thought $s'$, that $s$ can be object of my thought, itself an element of $S$. If we regard this as transform $\phi(s)$ of the element $s$ then has the transformation $\phi$ of $S$, thus determined, the property that the transform $S'$ is part of $S$; and $S'$ is certainly proper part of $S$, because there are elements in $S$ (e.~g., my own ego) which are different from such thought $s'$ and therefore are not contained in $S'$. Finally it is clear that if $a$, $b$ are different elements of $S$, their transforms $a'$, $b'$ are also different, that therefore the transformation $\phi$ is a distinct (similar) transformation (26). Hence $S$ is infinite, which was to be proved. \mypara{67.} Theorem. If $R$, $S$ are similar systems, then is $R$ finite or infinite according as $S$ is finite or infinite. Proof. If $S$ is infinite, therefore similar to a proper part $S'$ of itself, then if $R$ and $S$ are similar, $S'$ by (33) must be similar to $R$ and by (35) likewise similar to a proper part of $R$, which therefore by (33) is itself similar to $R$; therefore $R$ is infinite, which was to be proved. \mypara{68.} Theorem. Every system $S$, which possesses an infinite part is likewise infinite; or, in other words, every part of a finite system is finite. % [File: 070.png---\michael.j.white\\sahobart\glimpseofchaos\wwoods\----- Proof. If $T$ is infinite and there is hence such a similar transformation $\psi$ of $T$, that $\psi(T)$ is a proper part of $T$, then, if $T$ is part of $S$, we can extend this transformation $\psi$ to a transformation $\phi$ of $S$ in which, if $s$ denotes any element of $S$, we put $\phi(s)=\psi(s)$ or $\phi(s)=s$ according as $s$ is element of $T$ or not. This transformation $\phi$ is a similar one; for, if $a$, $b$ denote different elements of $S$, then if both are contained in $T$, the transform $\phi(a)=\psi(a)$ is different from the transform $\phi(b)=\psi(b)$, because $\psi$ is a similar transformation; if further $a$ is contained in $T$, but $b$ not, then is $\phi(a)=\psi(a)$ different from $\phi(b)=b$, because $\psi(a)$ is contained in $T$; if finally neither $a$ nor $b$ is contained in $T$ then also is $\phi(a)=a$ different from $\phi(b)=b$, which was to be shown. Since further $\psi(T)$ is part of $T$, because by (7) also part of $S$, it is clear that also $\phi(S) \partof S$. Since finally $\psi(T)$ is proper part of $T$ there exists in $T$ and therefore also in $S$, an element $t$, not contained in $\psi(T)=\phi(T)$; since then the transform $\phi(s)$ of every element $s$ not contained in $T$ is equal to $s$, and hence is different from $t$, $t$ cannot be contained in $\phi(S)$; hence $\phi(S)$ is proper part of $S$ and consequently $S$ is infinite, which was to be proved. \mypara{69.} Theorem. Every system which is similar to a part of a finite system, is itself finite. The proof follows from (67), (68). \mypara{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. % [File: 071.png] Proof. We have by (64) to show that if $\phi$ denotes any similar transformation of $S$ in itself, the transform $\phi(S)$ or $S'$ is never a proper part of $S$ but always $=S$. Obviously $S=\mathfrak{M}(a,T)$ and hence by (23), if the transforms are again denoted by accents, $S'=\mathfrak{M}(a',T')$, and, on account of the similarity of the transformation $\phi$, $a'$ is not contained in $T'$ (26). Since further by hypothesis $S' \partof S$, then must $a'$ and likewise every element of $T'$ either $=a$, or be element of $T$. If then---a case which we will treat first---$a$ is not contained in $T'$, then must $T' \partof T$ and hence $T'=T$, because $\phi$ is a similar transformation and because $T$ is a finite system; and since $a'$, as remarked, is not contained in $T'$, i.~e., not in $T$, then must $a'=a$, and hence in this case we actually have $S'=S$ as was stated. In the opposite case when $a$ is contained in $T'$ and hence is 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 $T=\mathfrak{M}(b,U)$ and by (15) $S=\mathfrak{M}(a,b,U)$, hence $S'=\mathfrak{M}(a',a,U')$. We now determine a new transformation $\psi$ of $T$ in which we put $\psi(b)=a'$, and generally $\psi(u)=u'$, whence by (23) $\psi(T)=\mathfrak{M}(a',U')$. Obviously $\psi$ is a similar transformation, because $\phi$ was such, and because $a$ is not contained in $U$ and therefore also $a'$ not in $U'$. Since further $a$ and every element $u$ is different from $b$ then (on account of the similarity of $\phi$) must also $a'$ and every element $u'$ be different from $a$ and consequently contained in $T$; hence $\psi(T) \partof T$ % [File: 072.png] and since $T$ is finite, therefore must $\psi(T) =T$, and $\mathfrak{M}(a',U')= T$. From this by (15) we obtain \[ \mathfrak{M}(a', a, U') = \mathfrak{M}(a, T) \] i.~e., according to the preceding $S' = S$. Therefore in this case also the proof demanded has been secured. %VI. \mysect{XII}{VI}{Simply Infinite Systems. Series of Natural Numbers} \mypara{71.} Definition. A system $N$ is said to be \textit{simply infinite} when there exists a similar transformation $\phi$ of $N$ in itself such that $N$ appears as chain (44) of an element not contained in $\phi(N)$. We call this element, which we shall denote in what follows by the symbol 1, the \textit{base-element} of $N$ and say the simply infinite system $N$ is \textit{set in order} [\textit{geordnet}] by this transformation $\phi$. If we retain the earlier convenient symbols for transforms and chains (IV) then the essence of a simply infinite system $N$ consists in the existence of a transformation $\phi$ of $N$ and an element 1 which satisfy the following conditions $\alpha$, $\beta$, $\gamma$, $\delta$: \begin{itemize} \item[] $\alpha$. $N' \partof N$. \item[] $\beta$. $N=1_0$. \item[] $\gamma$. The element 1 is not contained in $N'$. \item[] $\delta$. The transformation $\phi$ is similar. \end{itemize} Obviously it follows from $\alpha$, $\gamma$, $\delta$ that every simply infinite system $N$ is actually an infinite system (64) because it is similar to a proper part $N'$ of itself. % [File: 073.png] \mypara{72.} Theorem. In every infinite system $S$ a simply infinite system $N$ is contained as a part. Proof. By (64) there exists a similar transformation $\phi$ of $S$ such that $\phi(S)$ or $S'$ is a proper part of $S$; hence there exists an element 1 in $S$ which is not contained in $S'$. The chain $N=1_0$, which corresponds to this transformation $\phi$ of the system $S$ in itself (44), is a simply infinite system set in order by $\phi$; for the characteristic conditions $\alpha, \beta, \gamma, \delta$ in (71) are obviously all fulfilled. \mypara{73.} Definition. If in the consideration of a simply infinite system $N$ set in order by a transformation $\phi$ we entirely neglect the special character of the elements; simply retaining their distinguishability and taking into account only the relations to one another in which they are placed by the order-setting transformation $\phi$, then are these elements called \textit{natural numbers} or \textit{ordinal numbers} or simply \textit{numbers}, and the base-element 1 is called the \textit{base-number} of the \textit{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 $\alpha, \beta, \gamma, \delta$ in (71) and therefore are always the same in all ordered simply infinite systems, whatever names may happen to be given to the individual elements (compare 134), form the first object of the \textit{science of numbers} or \textit{arithmetic}. From the general notions and theorems of IV. about the transformation % [File: 074.png] of a system in itself we obtain immediately the following fundamental laws where $a$, $b$,~\ldots\ $m$, $n$,~\ldots\ always denote elements of $N$, therefore numbers, $A$, $B$, $C$,~\ldots\ parts of $N$, $a'$, $b'$,~\ldots\ $m'$, $n'$,~\ldots\ $A'$, $B'$, $C'$~\ldots\ the corresponding transforms, which are produced by the order-setting transformation $\phi$ and are always elements or parts of $N$; the transform $n'$ of a number $n$ is also called the number \textit{following} $n$. \mypara{74.} Theorem. Every number $n$ by (45) is contained in its chain $n_0$ and by (53) the condition $n \partof m_0$ is equivalent to $n_0 \partof m_0$. \mypara{75.} Theorem. By (57) $n'_0=(n_0)'=(n')_0$. \mypara{76.} Theorem. By (46) $n'_0 \partof n_0$. \mypara{77.} Theorem. By (58) $n_0 = \mathfrak{M}(n,n'_0)$. \mypara{78.} Theorem. $N = \mathfrak{M}(1,N')$, hence every number different from the base-number 1 is element of $N'$, i.~e., transform of a number. The proof follows from (77) and (71). \mypara{79.} Theorem. $N$ is the only number-chain containing the base-number 1. Proof. For if 1 is element of a number-chain $K$, then by (47) the associated chain $N \partof K$, hence $N=K$, because it is self-evident that $K \partof N$. \mypara{80.} Theorem of complete induction (inference from $n$ to $n'$). In order to show that a theorem holds for all numbers $n$ of a chain $m_0$, it is sufficient to show, $\rho$. that it holds for $n=m$, and $\sigma$. that from the validity of the theorem for a number % [File: 075.png] $n$ of the chain $m_0$ its validity for the following number $n'$ always follows. This results immediately from the more general theorem (59) or (60). The most frequently occurring case is where $m = 1$ and therefore $m_0$ is the complete number-series $N$. %VII. \mysect{XI}{VII}{Greater and Less Numbers} \mypara{81.} Theorem. Every number $n$ is different from the following number $n'$. Proof by complete induction (80): $\rho$. The theorem is true for the number $n = 1$, because it is not contained in $N'$ (71), while the following number $1'$ as transform of the number 1 contained in $N$ is element of $N'$. $\sigma$. If the theorem is true for a number $n$ and we put the following number $n' = p$, then is $n$ different from $p$, whence by (26) on account of the similarity (71) of the order-setting transformation $\phi$ it follows that $n'$, and therefore $p$, is different from $p'$. Hence the theorem holds also for the number $p$ following $n$, which was to be proved. \mypara{82.} Theorem. In the transform-chain $n'_0$ of a number $n$ by (74), (75) is contained its transform $n'$, but not the number $n$ itself. Proof by complete induction (80): $\rho$. The theorem is true for $n = 1$, because $1'_0 = N'$, % [File: 076.png] and because by (71) the base-number 1 is not contained in $N'$. $\sigma$. If the theorem is true for a number $n$, and we again put $n' =p$, then is $n$ not contained in $p_0$, therefore is it different from every number $q$ contained in $p_0$, whence by reason of the similarity of $\phi$ it follows that $n'$, and therefore $p$, is different from every number $q'$ contained in $p'_0$, and is hence not contained in $p'_0$. Therefore the theorem holds also for the number $p$ following $n$, which was to be proved. \mypara{83.} Theorem. The transform-chain $n'_0$ is proper part of the chain $n_0$. The proof follows from (76), (74), (82). \mypara{84.} Theorem. From $m_0= n_0$ it follows that $m=n$. Proof. Since by (74) $m$ is contained in $m_0$, and \[ m_0= n_0 = \mathfrak{M}(n, n'_0) \] by (77), then if the theorem were false and hence $m$ different from $n$, $m$ would be contained in the chain $n'_0$, hence by (74) also $m_0 \partof n'_0$, i.~e., $n_0 \partof n'_0$; but this contradicts theorem (83). Hence our theorem is established. \mypara{85.} Theorem. If the number $n$ is not contained in the number-chain $K$, then is $K \partof n'_0$. Proof by complete induction (80): $\rho$. By (78) the theorem is true for $n=1$. $\sigma$. If the theorem is true for a number $n$, then is it also true for the following number, $p= n'$; for if $p$ is not contained in the number-chain $K$, then by (40) $n$ also cannot be contained in $K$ and hence by our % [File: 077.png] hypothesis $K \partof n'_0$; now since by (77) $n'_0 = p_0 = \mathfrak{M}(p,p'_0)$, hence $K \partof \mathfrak{M}(p,p'_0)$ and $p$ is not contained in $K$, then must $K \partof p'_0$, which was to be proved. \mypara{86.} Theorem. If the number $n$ is not contained in the number-chain $K$, but its transform $n'$ is, then $K=n'_0$. Proof. Since $n$ is not contained in $K$, then by (85) $K \partof n'_0$, and since $n' \partof K$, then by (47) is also $n'_0 \partof K$, and hence $K=n'_0$, which was to be proved. \mypara{87.} Theorem. In every number-chain $K$ there exists one, and by (84) only one, number $k$, whose chain $k_0 = K$. Proof. If the base-number 1 is contained in $K$, then by (79) $K=N=1_0$. In the opposite case let $Z$ be the system of all numbers not contained in $K$; since the base-number 1 is contained in $Z$, but $Z$ is only a proper part of the number-series $N$, then by (79) $Z$ cannot be a chain, i.~e., $Z'$ cannot be part of $Z$; hence there exists in $Z$ a number $n$, whose transform $n'$ is not contained in $Z$, and is therefore certainly contained in $K$; since further $n$ is contained in $Z$, and therefore not in $K$, then by (86) $K=n'_0$, and hence $k = n'$, which was to be proved. \mypara{88.} Theorem. If $m$, $n$ are different numbers then by (83), (84) one and only one of the chains $m_0$, $n_0$ is proper part of the other and either $n_0 \partof m'_0$ or $m_0 \partof n'_0$. Proof. If $n$ is contained in $m_0$, and hence by (74) also $n_0 \partof m_0$, then $m$ can not be contained in the chain $n_0$ (because otherwise by (74) we should have $m_0 \partof n_0$, % [File: 078.png] therefore $m_0 = n_0$, and hence by (84) also $m = n$) and thence it follows by (85) that $n_0 \partof m'_0$. In the contrary case, when $n$ is not contained in the chain $m_0$, we must have by (85) $m_0 \partof n'_0$, which was to be proved. \mypara{89.} Definition. The number $m$ is said to be \textit{less} than the number $n$ and at the same time $n$ \textit{greater} than $m$, in symbols \[ m < n, \quad n >m, \] when the condition \[ n_0 \partof m'_0 \] is fulfilled, which by (74) may also be expressed \[ n \partof m'_0. \] \mypara{90.} Theorem. If $m$, $n$ are any numbers, then always one and only one of the following cases $\lambda$, $\mu$, $\nu$ occurs: \begin{eqnarray*} \lambda.& m = n,\ n = m,\ \mathrm{i.~e.}, &m_0 = n_0\\ \mu. & m < n,\ n > m,\ \mathrm{i.~e.}, &n_0 \partof m'_0\\ \nu. & m > n,\ n < m,\ \mathrm{i.~e.}, &m_0 \partof n'_0. \end{eqnarray*} Proof. For if $\lambda$ occurs (84) then can neither $\mu$ nor $\nu$ occur because by (83) we never have $n_0 \partof n'_0$. But if $\lambda$ does not occur then by (88) one and only one of the cases $\mu$, $\nu$ occurs, which was to be proved. \mypara{91.} Theorem. $n n$ (90) we use the symbols \[ m\leqq n \mathrm{\ or\ also\ } n\geqq m \] % [File: 079.png] and we say $m$ is \textit{at most equal} to $n$, and $n$ is \textit{at least equal} to $m$. \mypara{93.} Theorem. Each of the conditions \[ m \leqq n,\quad mm$. Hence the condition $m \leqq n$ is equivalent to $n_0 \partof m_0$. Besides it follows from (22), (27), (75) that this condition $n_0 \partof m_0$ is again equivalent to $n'_0 \partof m'_0$, i.~e., by $\mu$ in (90) to $mk$ (89), whence at once from (90) it follows that there exists in $T$ one and only one least number, which was to be proved. \mypara{97.} Theorem. The least number of the chain $n_0$ is $n$, and the base-number 1 is the least of all numbers. Proof. For by (74), (93) the condition $m\partof n_0$ is equivalent to $m\geqq n$. Or our theorem also follows immediately from the proof of the preceding theorem, because if in that we assume $T=n_0$, evidently $k=n$ (51). \mypara{98.} Definition. If $n$ is any number, then will we denote by $Z_n$ the system of all numbers that are \emph{not greater} than $n$, and hence \emph{not} contained in $n'_0$. The condition \[ m\partof Z_n \] by (92), (93) is obviously equivalent to each of the following conditions: \[ m\leqq n,\quad mm,\quad Z_{m'}\partof Z_n\\ \nu. & \quad m>n,\quad nm$, therefore $Z_m$ is proper part of $Z_n$. Conversely if $Z_m$ is proper part of $Z_n$ then by (100) $m\leqq n$, and since $m$ cannot be $=n$, because otherwise $Z_m=Z_n$, we must have $mn$, hence by (94) also $m'>n'$ and consequently by (98) $m'$ would not be contained in $Z_{n'}$); but from $m \partof Z_n$ we have $m' \partof Z'_n$, and hence certainly \[ Z_{n'} \partof \mathfrak{M}(1, Z'_n). \] Since conversely by (99) $1 \partof Z_n$, and by (109) $Z'_n \partof Z_{n'}$, then by (10) we have $\mathfrak{M}(1, Z'_n) \partof Z_{n'}$ and hence our theorem follows by (5). \mypara{111.} Definition. If in a system $E$ of numbers there exists an element $g$, which is greater than every other number contained in $E$, then $g$ is said to be the \textit{greatest} number of the system $E$, and by (90) there can evidently be only one such greatest number in $E$. If a system consists of a single number, then is this number itself the greatest number of the system. \mypara{112.} Theorem. By (98) $n$ is the greatest number of the system $Z_n$. \mypara{113.} Theorem. If there exists in $E$ a greatest number $g$, then is $E \partof Z_g$. Proof. For every number contained in $E$ is $\leqq g$, % [File: 084.png] and hence by (98) contained in $Z_g$, which was to be proved. \mypara{114.} Theorem. If $E$ is part of a system $Z_n$, or what amounts to the same thing, there exists a number $n$ such that all numbers contained in $E$ are $\leqq n$, then $E$ possesses a greatest number $g$. Proof. The system of all numbers $p$ satisfying the condition $E \partof Z_p$---and by our hypothesis such numbers exist---is a chain (37), because by (107), (7) it follows also that $E \partof Z_{p'}$, and hence by (87) $=g_0$, where $g$ signifies the least of these numbers (96), (97). Hence also $E \partof Z_g$, therefore by (98) every number contained in $E$ is $\leqq g$, and we have only to show that the number $g$ is itself contained in $E$. This is immediately obvious if $g=1$, for then by (102) $Z_g$, and consequently also $E$ consists of the single number 1. But if $g$ is different from 1 and consequently by (78) the transform $f'$ of a number $f$, then by (108) is $E \partof \mathfrak{M}(Z_f,g)$; if therefore $g$ were not contained in $E$, then would $E \partof Z_f$, and there would consequently be among the numbers $p$ a number $f$ by (91) $t$, 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. \mypara{118.} Theorem. In $N$ the number $n'$ is next greater than $n$, and $n$ next less than $n'$. The proof follows from (116), (117). % [File: 086.png] %VIII. \mysect{X}{VIII}{Finite and Infinite Parts of the Number-Series} \mypara{119.} Theorem. Every system $Z_n$ in (98) is finite. Proof by complete induction (80). $\rho$. By (65), (102) the theorem is true for $n=1$. $\sigma$. If $Z_n$ is finite, then from (108) and (70) it follows that $Z_{n'}$ is also finite, which was to be proved. \mypara{120.} Theorem. If $m$, $n$ are different numbers, then are $Z_m$, $Z_n$ dissimilar systems. Proof. By reason of the symmetry we may by (90) assume that $mn$, hence $m \partof n'_0$, and the theorem to prove comes to this that $\psi(n)$ is not contained in $\psi(n'_0)$, and hence by (127) is not contained in $\theta\psi(n_0)$. This we establish for every number $n$ by complete induction (80). In fact, $\rho$. \label{Ind132}this theorem is true by $\gamma$ for $n = 1$, since $\psi(1)=\omega$ and $\psi(1_0)= \psi(N)=\Omega$. $\sigma$. If the theorem is true for a number $n$, then is it also true for the following number $n'$; for if $\psi(n')$, i.~e., $\theta \psi(n)$, were contained in $\theta\psi(n'_0)$, then by $\delta$ and (27), $\psi(n)$ would also be contained in $\psi(n'_0)$ while our hypothesis states just the opposite; which was to be proved. \mypara{133.} Theorem. Every system which is similar to a simply infinite system and therefore by (132), (33) to the number-series $N$ is simply infinite. Proof. If $\Omega$ is a system similar to the number-series $N$, then by (32) there exists a similar transformation $\psi$ of $N$ such that % [File: 099.png] \begin{align*} & \text{I\@. } \psi(N) = \Omega; \shortintertext{then we put } & \text{II\@. } \psi(1) = \omega. \end{align*} If we denote, as in (26), by $\overline{\psi}$ the inverse, likewise similar transformation of $\Omega$, then to every element $\nu$ of $\Omega$ there corresponds a determinate number $\overline{\psi}(\nu)=n$, viz., that number whose transform $\psi(n) = \nu$. Since to this number $n$ there corresponds a determinate following number $\phi(n)=n'$, and to this again a determinate element $\psi(n')$ in $\Omega$ there belongs to every element $\nu$ of the system $\Omega$ a determinate element $\psi(n')$ of that system which as transform of $\nu$ we shall designate by $\theta(\nu)$. Thus a transformation $\theta$ of $\Omega$ in itself is completely determined,\footnote {Evidently $\theta$ is the transformation $\psi\,\phi\,\overline{\psi}$ compounded by (25) out of $\overline{\psi}$, $\phi$, $\psi$.} and in order to prove our theorem we will show that by $\theta \Omega$ is set in order (71) as a simply infinite system, i.~e., that the conditions $\alpha$, $\beta$, $\gamma$, $\delta$ stated in the proof of (132) are all fulfilled. First $\alpha$ is immediately obvious from the definition of $\theta$. Since further to every number $n$ corresponds an element $\nu = \phi(n)$, for which $\theta(\nu) = \psi(n')$, we have generally, \[ \mathrm{III.\ }\psi(n') = \theta\psi(n), \] and thence in connection with I, II, $\alpha$ it results that the transformations $\theta$, $\psi$ fulfill all the conditions of theorem (126); therefore $\beta$ follows from (128) and I\@. Further by (127) and I \[ \psi(N') = \theta \psi(N) = \theta(\Omega), \] and thence in combination with II and the similarity % [File: 100.png] of the transformation $\psi$ follows $\gamma$, because otherwise $\psi(1)$ must be contained in $\psi(N')$, hence by (27) the number 1 in $N'$, which by (71, $\gamma$) is not the case. If finally $\mu$, $\nu$ denote elements of $\Omega$ and $m$, $n$ the corresponding numbers whose transforms are $\psi(m) = \mu$, $\psi(n) = \nu$, then from the hypothesis $\theta(\mu) = \theta(\nu)$ it follows by the foregoing that $\psi(m') = \psi(n')$, thence on account of the similarity of $\psi$, $\phi$ that $m' = n'$, $m = n$, therefore also $\mu = \nu$; hence also $\delta$ is true, which was to be proved. \mypara{134.} Remark. By the two preceding theorems (132), (133) all simply infinite systems form a class in the sense of (34). At the same time, with reference to (71), (73) it is clear that every theorem regarding numbers, i.~e., regarding the elements $n$ of the simply infinite system $N$ set in order by the transformation $\phi$, and indeed every theorem in which we leave entirely out of consideration the special character of the elements $n$ and discuss only such notions as arise from the arrangement $\phi$, possesses perfectly general validity for every other simply infinite system $\Omega$ set in order by a transformation $\theta$ and its elements $\nu$, and that the passage from $N$ to $\Omega$ (e.~g., also the translation of an arithmetic theorem from one language into another) is effected by the transformation $\psi$ considered in (132), (133), which changes every element $n$ of $N$ into an element $\nu$ of $\Omega$, i.~e., into $\psi(n)$. This element $\nu$ can be called the $n$th element of $\Omega$ and accordingly the number $n$ is itself the $n$th number of the number-series % [File: 101.png] $N$. The same significance which the transformation $\phi$ possesses for the laws in the domain $N$, in so far as every element $n$ is followed by a determinate element $\phi(n)=n'$, is found, after the change effected by $\psi$, to belong to the transformation $\theta$ for the same laws in the domain $\Omega$, in so far as the element $\nu=\psi(n)$ arising from the change of $n$ is followed by the element $\theta(\nu)=\psi(n')$ arising from the change of $n'$; we are therefore justified in saying that by $\psi \phi$ is changed into $\theta$, which is symbolically expressed by $\theta=\psi\phi\overline{\psi}$, $\phi=\overline{\psi}\theta\psi$. By these remarks, as I believe, the definition of the notion of numbers given in (73) is fully justified. We now proceed to further applications of theorem (126). %XI. \mysect{VII}{XI}{Addition of Numbers} \mypara{135.} Definition. It is natural to apply the definition set forth in theorem (126) of a transformation $\psi$ of the number-series $N$, or of the \textit{function} $\psi(n)$ determined by it to the case, where the system there denoted by $\Omega$ in which the transform $\psi(N)$ is to be contained, is the number-series $N$ itself, because for this system $\Omega$ a transformation $\theta$ of $\Omega$ in itself already exists, viz., that transformation $\phi$ by which $N$ is set in order as a simply infinite system (71), (73). Then is also $\Omega=N$, $\theta(n)=\phi(n)=n'$, hence \[ \mathrm{I.\ }\psi(N) \partof N, \] and it remains in order to determine $\psi$ completely % [File: 102.png] only to select the element $\omega$ from $\Omega$, i.~e., from $N$, at pleasure. If we take $\omega=1$, then evidently $\psi$ becomes the identical transformation (21) of $N$, because the conditions \[ \psi(1)=1,\quad \psi(n')=(\psi(n))' \] are generally satisfied by $\psi(n)=n$. If then we are to produce another transformation $\psi$ of $N$, then for $\omega$ we must select a number $m'$ different from 1, by (78) contained in $N$, where $m$ itself denotes any number; since the transformation $\psi$ is obviously dependent upon the choice of this number $m$, we denote the corresponding transform $\psi(n)$ of an arbitrary number $n$ by the symbol $m+n$, and call this number the \textit{sum} which arises from the number $m$ by the \textit{addition} of the number $n$, or in short the sum of the numbers $m$, $n$. Therefore by (126) this sum is completely determined by the conditions\footnote{The above definition of addition based immediately upon theorem (126) seems to me to be the simplest. By the aid of the notion developed in (131) we can, however, define the sum $m+n$ by $\phi^n(m)$ or also by $\phi^m(n)$, where $\phi$ has again the foregoing meaning. In order to show the complete agreement of these definitions with the foregoing, we need by (126) only to show that if $\phi^n(m)$ or $\phi^m(n)$ is denoted by $\psi(n)$, the conditions $\psi(1)=m'$, $\psi(n')=\phi\psi(n)$ are fulfilled which is easily done with the aid of complete induction (80) by the help of (131).} \begin{align*} \text{II\@. }& m+1=m',\\ \text{III\@. }& m+n'=(m+n)'. \end{align*} \mypara{136.} Theorem. $m'+n=m+n'$. Proof by complete induction (80). For $\rho$. the theorem is true for $n=1$, since by (135, II) \[ m'+1=(m')'=(m+1)', \] and by (135, III) $(m+1)'=m+1'$. % [File: 103.png] $\sigma$. If the theorem is true for a number $n$, and we put the following number $n'=p$ then is $m'+n= m+p$, hence also $(m'+n)'=(m+p)'$, whence by (135, III) $m'+p=m+p'$; therefore the theorem is true also for the following number $p$, which was to be proved. \mypara{137.} Theorem. $m'+n=(m+n)'$. The proof follows from (136) and (135, III). \mypara{138.} Theorem. $1+n=n'$. Proof by complete induction (80). For $\rho$. by (135, II) the theorem is true for $n=1$. $\sigma$. If the theorem is true for a number $n$ and we put $n'=p$, then $1+n=p$, therefore also $(1+n)'=p'$, whence by (135, III) $1+p=p'$, i.~e., the theorem is true also for the following number $p$, which was to be proved. \mypara{139.} Theorem. $1+n=n+1$. The proof follows from (138) and (135, II). \mypara{140.} Theorem. $m+n=n+m$. Proof by complete induction (80). For $\rho$. by (139) the theorem is true for $n=1$. $\sigma$. If the theorem is true for a number $n$, then there follows also $(m+n)'=(n+m)'$, i.~e., by (135, III) $m+n'=n+m'$, hence by (136) $m+n'=n'+m$; therefore the theorem is also true for the following number $n'$, which was to be proved. \mypara{141.} Theorem. $(l+m)+n=l+(m+n)$. Proof by complete induction (80). For % [File: 104.png] $\rho$. the theorem is true for $n=1$, because by (135, II, III, II) $(l+m)+1=(l+m)'=l+m'=l+(m+1)$. $\sigma$. If the theorem is true for a number $n$, then there follows also $\bigl((l+m)+n\bigr)'=\bigl(l+(m+n)\bigr)'$, i.~e., by (135, III) \[ (l+m)+n'=l+(m+n)'=l+(m+n'), \] therefore the theorem is also true for the following number $n'$, which was to be proved. \mypara{142.} Theorem. $m+n>m$. Proof by complete induction (80). For $\rho$. by (135, II) and (91) the theorem is true for $n = 1$. $\sigma$. If the theorem is true for a number $n$, then by (95) it is also true for the following number $n'$, because by (135, III) and (91) \[ m+n'=(m+n)'>m+n, \] which was to be proved. \mypara{143.} Theorem. The conditions $m>a$ and $m+n> a+n$ are equivalent. Proof by complete induction (80). For $\rho$. by (135, II) and (94) the theorem is true for $n = 1$. $\sigma$. If the theorem is true for a number $n$, then is it also true for the following number $n'$, since by (94) the condition $m+n>a+n$ is equivalent to $(m+n)'> (a+n)'$, hence by (135, III) also equivalent to \[ m+n'> a+n', \] which was to be proved. % [File: 105.png] \mypara{144.} Theorem. If $m>a$ and $n>b$, then is also \[ m+n>a+b. \] Proof. For from our hypotheses we have by (143) $m+n>a+n$ and $n+a>b+a$ or, what by (140) is the same, $a+n>a+b$, whence the theorem follows by (95). \mypara{145.} Theorem. If $m+n=a+n$ then $m=a$. Proof. For if $m$ does not $=a$, hence by (90) either $m>a$ or $m a+n$ or $m+nn$, then there exists one and by (157) only one number $m$ which satisfies the condition $m+n=l$. Proof by complete induction (80). For $\rho$. the theorem is true for $n = 1$. In fact, if $l>1$, i.~e., (89) if $l$ is contained in $N'$, and hence is the transform $m'$ of a number $m$, then by (135, II) it follows that $l=m+1$, which was to be proved. $\sigma$. If the theorem is true for a number $n$, then we show that it is also true for the following number $n'$. In fact, if $l>n'$, then by (91), (95) also $l>n$, and hence there exists a number $k$ which satisfies the condition $l=k+n$; since by (138) this is different from 1 (otherwise $l$ would be $=n'$) 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. % [File: 106.png] %XII. \mysect{VI}{XII}{Multiplication of Numbers} \mypara{147.} Definition. After having found in XI an infinite system of new transformations of the number-series $N$ in itself, we can by (126) use each of these in order to produce new transformations $\psi$ of $N$. When we take $\Omega = N$, and $\theta(n) = m+n = n+m$, where $m$ is a determinate number, we certainly again have \[ \text{I\@. } \psi(N) \partof N, \] and it remains, to determine $\psi$ completely only to select the element $\omega$ from $N$ at pleasure. The simplest case occurs when we bring this choice into a certain agreement with the choice of $\theta$, by putting $\omega = m$. Since the thus perfectly determinate $\psi$ depends upon this number $m$, we designate the corresponding transform $\psi(n)$ of any number $n$ by the symbol $m \times n$ or $m\centerdot n$ or $mn$, and call this number the \textit{product} arising from the number $m$ by \textit{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 \begin{align*} \text{II\@. }& m\centerdot 1=m\\ \text{III\@. }& mn'=mn+m \end{align*} \mypara{148.} Theorem. $m'n = mn+n$. Proof by complete induction (80). For $\rho$. by (147, II) and (135, II) the theorem is true for $n = 1$. % [File: 107.png] $\sigma$. If the theorem is true for a number $n$, we have \[ m'n+m'=(mn+n)+m' \] and consequently by (147, III), (141), (140), (136), (141), (147, III) \[ m'n'=mn+(n+m')=mn+(m'+n)=mn+(m+n') \] \[ =(mn+m)+n'=mn'+n'; \] therefore the theorem is true for the following number $n'$, which was to be proved. \mypara{149.} Theorem. $1\centerdot n = n$. Proof by complete induction (80). For $\rho$. by (147, II) the theorem is true for $n = 1$. $\sigma$. If the theorem is true for a number $n$, then we have $1\centerdot n+1=n+1$, i.~e., by (147, III), (135, II) $1\centerdot n'=n'$, therefore the theorem also holds for the following number $n'$, which was to be proved. \mypara{150.} Theorem. $mn=nm$. Proof by complete induction (80). For $\rho$. by (147, II), (149) the theorem is true for $n = 1$. $\sigma$. 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 theorem is also true for the following number $n'$, which was to be proved. \mypara{151.} Theorem. $l(m+n)=lm+ln$. Proof by complete induction (80). For $\rho$. by (135, II), (147, III), (147, II) the theorem is true for $n = 1$. % [File: 108.png] $\sigma$. If the theorem is true for a number $n$, we have \[ l(m+n)+l=(lm+ln)+l; \] but by (147, III), (135, III) we have \[ l(m+n)+l=l(m+n)'=l(m+n'), \] and by (141), (147, III) \[ (lm+ln)+l=lm+(ln+l)=lm+ln', \] consequently $l(m+n')=lm+ln'$, i.~e., the theorem is true also for the following number $n'$, which was to be proved. \mypara{152.} Theorem. $(m+n)l=ml+nl$. The proof follows from (151), (150). \mypara{153.} Theorem. $(lm)n=l(mn)$. Proof by complete induction (80). For $\rho$. by (147, II) the theorem is true for $n = 1$. $\sigma$. If the theorem is true for a number $n$, then we have \[ (lm)n+lm=l(mn)+lm, \] i.~e., by (147, III), (151), (147, III) \[ (lm)n'=l(mn+m)=l(mn'), \] hence the theorem is also true for the following number $n'$, which was to be proved. \mypara{154.} Remark. If in (147) we had assumed no relation between $\omega$ and $\theta$, but had put $\omega=k$, $\theta(n)= m+n$, then by (126) we should have had a less simple transformation $\psi$ of the number-series $N$; for the number~1 would $\psi(1)=k$ and for every other number (therefore contained in the form $n'$) would $\psi(n')= mn+k$; since thus would be fulfilled, as one could % [File: 109.png] easily convince himself by the aid of the foregoing theorems, the condition $\psi(n')=\theta\psi(n)$, i.~e., $\psi(n')= m+\psi(n)$ for all numbers $n$. %XIII. \mysect{V}{XIII}{Involution of Numbers} \mypara{155.} Definition. If in theorem (126) we again put $\Omega = N$, and further $\omega=a$, $\theta(n)=an=na$, we get a transformation $\psi$ of $N$ which still satisfies the condition \[ \text{I\@. }\psi(N) \partof N; \] the corresponding transform $\psi(n)$ of any number $n$ we denote by the symbol $a^n$, and call this number a \textit{power of the base} $a$, while $n$ is called the \textit{exponent} of this power of $a$. Hence this notion is completely determined by the conditions \begin{align*} \text{II\@. }& a^1=a\\ \text{III\@. }& a^{n'}=a\centerdot a^n =\rlap{$a^n\centerdot a$}. \end{align*} \mypara{156.} Theorem. $a^{m+n}=a^m\centerdot a^n$. Proof by complete induction (80). For $\rho$. by (135, II), (155, III), (155, II) the theorem is true for $n = 1$. $\sigma$. If the theorem is true for a number $n$, we have \[ a^{m+n}\centerdot a=(a^m\centerdot a^n)a; \] but by (155, III), (135, III) $a^{m+n}\centerdot a=a^{(m+n)'}=a^{m+n'}$, and by (153), (155, III) $(a^m\centerdot a^n)a=a^m(a^n\centerdot a)=a^m\centerdot a^{n'}$; hence $a^{m+n'}=a^m\centerdot a^{n'}$, i.~e., the theorem is also true for the following number $n'$, which was to be proved. \mypara{157.} Theorem. $(a^m)^n=a^{mn}$. % [File: 110.png] Proof by complete induction (80). For $\rho$. by (155, II), (147, II) the theorem is true for $n = 1$. $\sigma$. If the theorem is true for a number $n$, we have \[ (a^m)^n\centerdot a^m=a^{mn}\centerdot a^m \] but by (155, III) $(a^m)^n\centerdot a^m=(a^m)^{n'}$, and by (156), (147, III) $a^{mn}\centerdot a^m=a^{mn+m}=a^{mn'}$; hence $(a^m)^{n'}=a^{mn'}$, i.~e., the theorem is also true for the following number $n'$, which was to be proved. \mypara{158.} Theorem. $(ab)^n=a^n\centerdot b^n$ Proof by complete induction (80). For $\rho$. by (155, II) the theorem is true for $n = 1$. $\sigma$. If the theorem is true for a number $n$, then by (150), (153), (155, III) we have also $(ab)^n\centerdot a= a(a^n\centerdot b^n)=(a\centerdot a^n)b^n=a^{n'}\centerdot b^n$, and thus $\bigl( (ab)^n\centerdot a \bigr)b= (a^{n'}\centerdot b^n)b$; but by (153), (155, III) $\bigl( (ab)^n\centerdot a \bigr)b= (ab)^n\centerdot (ab)=(ab)^{n'}$, and likewise \[ (a^{n'}\centerdot b^n)b = a^{n'}\centerdot (b^n\centerdot b) = a^{n'}\centerdot b^{n'}; \] therefore $(ab)^{n'}=a^{n'}\centerdot b^{n'}$ i.~e., the theorem is also true for the following number $n'$, which was to be proved. %XIV. \mysect{II}{XIV}{Number of the Elements of a Finite System} \mypara{159.} Theorem. If $\Sigma$ is an infinite system, then is every one of the number-systems $Z_n$ defined in (98) similarly transformable in $\Sigma$ (i.~e., similar to a part of $\Sigma$), and conversely. Proof. If $\Sigma$ is infinite, then by (72) there certainly exists a part $T$ of $\Sigma$, which is simply infinite, therefore % [File: 111.png] by (132) similar to the number-series $N$, and consequently by (35) every system $Z_n$ as part of $N$ is similar to a part of $T$, therefore also to a part of $\Sigma$, which was to be proved. The proof of the converse---however obvious it may appear---is more complicated. If every system $Z_n$ is similarly transformable in $\Sigma$, then to every number $n$ corresponds such a similar transformation $\alpha_n$ of $Z_n$ that $\alpha_n(Z_n) \partof \Sigma$. From the existence of such a series of transformations $\alpha_n$, regarded as given, but respecting which nothing further is assumed, we derive first by the aid of theorem (126) the existence of a new series of such transformations $\psi_n$ possessing the special property that whenever $m \leqq n$, hence by (100) $Z_m \partof Z_n$, the transformation $\psi_m$ of the part $Z_m$ is contained in the transformation $\psi_n$ of $Z_n$ (21), i.~e., the transformations $\psi_m$ and $\psi_n$ completely coincide with each other for all numbers contained in $Z_m$, hence always \[ \psi_m(m) = \psi_n(m). \] In order to apply the theorem stated to gain this end we understand by $\Omega$ that system whose elements are all possible similar transformations of all systems $Z_n$ in $\Sigma$, and by aid of the given elements $\alpha_n$, likewise contained in $\Omega$, we define in the following manner a transformation $\theta$ of $\Omega$ in itself. If $\beta$ is any element of $\Omega$, thus, e.~g., a similar transformation of the determinate system $Z_n$ in $\Sigma$, then the system $\alpha_{n'}(Z_{n'})$ cannot be part of $\beta(Z_n)$, for otherwise $Z_{n'}$ would be % [File: 112.png] similar by (35) to a part of $Z_n$, hence by (107) to a proper part of itself, and consequently infinite, which would contradict theorem (119); therefore there certainly exists in $Z_{n'}$ one number or several numbers $p$ such that $\alpha_{n'}(p)$ is not contained in $\beta(Z_n)$; from these numbers $p$ we select---simply to lay down something determinate---always the least $k$ (96) and, since $Z_{n'}$ by (108) is compounded out of $Z_n$ and $n'$, define a transformation $\gamma$ of $Z_{n'}$ such that for all numbers $m$ contained in $Z_n$ the transform $\gamma(m)=\beta(m)$ and besides $\gamma(n')=\alpha_{n'}(k)$; this obviously similar transformation $\gamma$ of $Z_{n'}$ in $\Sigma$ we consider then as a transform $\theta(\beta)$ of the transformation $\beta$, %[**Greek: beta] and thus a transformation $\theta$ of the system $\Omega$ in itself is completely defined. After the things named $\Omega$ and $\theta$ in (126) are determined we select finally for the element of $\Omega$, denoted by $\omega$ the given transformation $\alpha_1$; thus by (126) there is determined a transformation $\psi$ of the number-series $N$ in $\Omega$, which, if we denote the transform belonging to an arbitrary number $n$, not by $\psi(n)$ but by $\psi_n$, satisfies the conditions \begin{align*} \text{II\@. }& \psi_1=\alpha_1\\ \text{III\@. }& \psi_{n'}=\theta(\psi_n) \end{align*} By complete induction (80) it results first that $\psi_n$ is a similar transformation of $Z_n$ in $\Sigma$; for $\rho$. by II this is true for $n =1$. $\sigma$. if this statement is true for a number $n$, it follows from III and from the character of the above described transition $\theta$ from $\beta$ to $\gamma$, that the statement is % [File: 113.png] also true for the following number $n'$, which was to be proved. Afterward we show likewise by complete induction (80) that if $m$ is any number the above stated property \[ \psi_n(m)=\psi_m(m) \] actually belongs to all numbers $n$, which are $\geqq m$, and therefore by (93), (74) belong to the chain $m_0$; in fact, $\rho$. this is immediately evident for $n = m$, and $\sigma$. if this property belongs to a number $n$ it follows again from III and the nature of $\theta$, that it also belongs to the number $n'$, which was to be proved. After this special property of our new series of transformations $\psi_n$ has been established, we can easily prove our theorem. We define a transformation $\chi$ of the number-series $N$, in which to every number $n$ we let the transform $\chi(n)=\psi_n(n)$ correspond; obviously by (21) all transformations $\psi_n$ are contained in this one transformation $\chi$. Since $\psi_n$ was a transformation of $Z_n$ in $\Sigma$, it follows first that the number-series $N$ is likewise transformed by $\chi$ in $\Sigma$, hence $\chi(N) \partof \Sigma$. 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