Dedekind: The Nature and Meaning of Numbers

Preface to the first edition.

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. [*] 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. [†] 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, [‡] must, in my judgment, the whole science of numbers be established. The design of such a presentation I had formed before the publication of my paper on Continuity, but only after its appearance and with many interruptions 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 Anzahl of things (161), the 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 [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 ἀεί ὁ ἄνθρωπος ἀριθμητίζει, I hope that the following pages, as an attempt to establish the science of numbers upon a uniform foundation will find a generous 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 $sqrt{2}cdotsqrt{3}=sqrt{6}$ 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.}mboxqquad 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.}mboxqquad 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, $a$, $b$, $c$, ldotsfor 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 $a$, $b$, $c$, ldotstextit{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 $A$, $B$, $C$,~ldotsto 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 $A$, $B$, $C$,~ldotsis carefully to be distinguished from the system whose elements are the systems $A$, $B$, $C$,~ldotsthemselves.

mypara{9.} Theorem. The systems $A$, $B$, $C$, ldotsare parts of $mathfrak{M}(A, B, C,,ldots)$.

The proof follows from (8), (3).

mypara{10.} Theorem. If $A$, $B$, $C$,~ldotsare 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 $A$, $B$, $C$,~ldotsthen is $P partof mathfrak{M}(A, B, C,,ldots)$.

The proof follows from (9), (7).

mypara{12.} Theorem. If each of the systems $P$, $Q$,~ldotsis part of one of the systems $A$, $B$, $C$,~ldotsthen 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 $P$, $Q$,~ldotsthen 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 $A$, $B$, $C$,~ldotsis compounded out of any of the systems $P$, $Q$,~ldotsthen 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 $P$, $Q$,~ldotsis 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{thenis} mathfrak{M}(A, R) = mathfrak{M}(P, B). end{gather*}

Proof. For by the preceding theorem (15) [ mathfrak{M}(A, R) mathrm{aswellas} 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 $A$, $B$, $C$,~ldotswhen $T$ is part of each of these systems; and by the textit{community} [textit{Gemeinheit}] of the systems $A$, $B$, $C$,~ldotswe understand the perfectly determinate system $mathfrak{G}(A, B, C,,ldots)$ which consists of all the common elements $g$ of $A$, $B$, $C$,~ldotsand % [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 $A$, $B$, $C$,~ldotspossess 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 $A$, $B$, $C$,~ldotsis 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 $A$, $B$, $C$,~ldotsis whole (3) of one of the systems $P$, $Q$,~ldotsthen 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 $A$, $B$, $C$,~ldotsand 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 $A$, $B$, $C$,~ldotsare by (9) parts of $M$, and hence $A'$, $B'$, $C'$,~ldotsby (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')=psibigl(phi(s)bigr),

] where again we have put $phi(s)=s'$. This transformation $theta$ can be denoted briefly by the symbol $psicenterdotphi$ or $psiphi$, the transform $theta(s)$ by $psiphi(s)$ where the order of the symbols $phi$, $psi$ is to be considered, since in general the symbol $phipsi$ has no interpretation and actually has meaning only when $psi(s') partof s$. If now $chi$ signifies a transformation of the system $psi(s')=psiphi(s)$ and $eta$ the transformation $chipsi$ of the system $S'$ compounded out of $psi$ and $chi$, then is $chitheta(s)=chipsi(s')=eta(s')=etaphi(s)$; therefore the compound transformations $chitheta$ and $etaphi$ coincide for every element $s$ of $S$, i.~e., $chitheta = eta phi$. In accordance with the meaning of $theta$ and $eta$ this theorem can finally be expressed in the form [

chicenterdotpsiphi = chipsicenterdotphi,

] % [File: 058.png] and this transformation compounded out of $phi$, $psi$, $chi$ can be denoted briefly by $chipsiphi$.

%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 $A'$, $B'$, $C'$,~ldotsthen by (27) must $g$ be common element of $A$, $B$, $C$,~ldotstherefore 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 $psiphi$ 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 $psi(a')=psiphi(a)$, $psi(b') =psiphi(b)$ and therefore $psiphi$ is a similar transformation. Besides every element $psiphi(s) = psi(s')$ of the system $psiphi(S)$ is transformed by $overline{psi}$ into $s' = phi(s)$ and % [File: 060.png] this by $overline{phi}$ into $s$, therefore $psiphi(s)$ 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) $psiphi$ is a similar transformation of $S$ such that $psiphi(S) = Q$, 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 $A$, $B$, $C$,~ldotsis 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.] $B' partof B$, $C' partof C$,~ldotstherefore 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.] $B$, $C$,~ldotsis 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 $A$, $B$, $C$,~ldotsand 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 $phibigl(phi_0(A)bigr) = phi_0bigl(phi(A)bigr)$.

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 $A$, $B$, $C$,~ldotsis 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 $A$, $B$, $C$,~ldotsis part of $M$, and hence by (45), (7) also part of the chain $M_0$, then by (47) must also each of the systems $A_0$, $B_0$, $C_0$,~ldotsbe 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 $A_0$, $B_0$, $C_0$,~ldotsby (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\sahobartglimpseofchaoswwoods-----

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\sahobartglimpseofchaoswwoods-----

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$m$, $n$,~ldotsalways denote elements of $N$, therefore numbers, $A$, $B$, $C$,~ldotsparts of $N$, $a'$, $b'$,~ldots$m'$, $n'$,~ldots$A'$, $B'$, $C'$~ldotsthe 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. $n<n'$.

Proof. For the condition for the case $nu$ in (90) is fulfilled by $m = n'$.

mypara{92.} Definition. To express that $m$ is either $=n$ or $< n$, hence not $> n$ (90) we use the symbols [ mleqq n mathrm{oralso} ngeqq 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 m<n',quad n_0 partof m_0 ] is equivalent to each of the others.

Proof. For if $m leqq n$, then from $lambda, mu$ in (90) we always have $n_0 partof m_0$, because by (76) $m'_0 partof m$. Conversely, if $n_0 partof m_0$, and therefore by (74) also $n partof m_0$, it follows from $m_0 = mathfrak{M}(m, m'_0)$ that either $n = m$, or $n partof m'_0$, i.~e., $n>m$. 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 $m<n'$, which was to be proved.

mypara{94.} Theorem. Each of the conditions [ m' leqq n,quad m'<n',quad m<n ] is equivalent to each of the others.

The proof follows immediately from (93), if we replace in it $m$ by $m'$, and from $mu$ in (90).

mypara{95.} Theorem. If $l<m$ and $mleqq n$ or if $l leqq m$, and $m<n$, then is $l<n$. But if $l leqq m$ and $m leqq n$, then is $l leqq n$.

Proof. For from the corresponding conditions (89), (93) $m_0 partof l'_0$ and $n_0 partof m_0$, we have by (7) $n_0 partof l'_0$ and the same thing comes also from the conditions $m_0 partof l_0$ and $n_0 partof m'_0$, because in consequence of the former we have also $m'_0 partof l'_0$. Finally from $m_0 partof l_0$ and $n_0 partof m_0$ we have also $n_0 partof l_0$ which was to be proved.

mypara{96.} Theorem. In every part $T$ of $N$ there exists one and only one textit{least} number $k$, i.~e., a number $k$ % [File: 080.png] which is less than every other number contained in $T$. If $T$ consists of a single number, then is it also the least number in $T$.

Proof. Since $T_0$ is a chain (44), then by (87) there exists one number $k$ whose chain $k_0=T_0$. Since from this it follows by (45), (77) that $Tpartofmathfrak{M}(k, k'_0)$, then first must $k$ itself be contained in $T$ (because otherwise $Tpartof k'_0$, hence by (47) also $T_0partof k'_0$, i.~e., $kpartof k'_0$, which by (83) is impossible), and besides every number of the system $T$, different from $k$, must be contained in $k'_0$, i.~e., be $>k$ (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 $mpartof n_0$ is equivalent to $mgeqq n$. 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 [ mpartof Z_n ] by (92), (93) is obviously equivalent to each of the following conditions: [ mleqq n,quad m<n',quad n_0partof m_0. ]

mypara{99.} Theorem. $1partof Z_n$ and $npartof Z_n$.

% [File: 081.png]

The proof follows from (98) or from (71) and (82).

mypara{100.} Theorem. Each of the conditions equivalent by (98) [ mpartof Z_n,mleqq n,m<n',n_0partof m_0 ] is also equivalent to the condition [ Z_mpartof Z_n. ]

Proof. For if $mpartof Z_n$, and hence $mleqq n$, and if $lpartof Z_m$, and hence $lleqq m$, then by (95) also $lleqq n$, i.~e., $lpartof Z_n$; if therefore $mpartof Z_n$, then is every element $l$ of the system $Z_m$ also element of $Z_n$, i.~e., $Z_mpartof Z_n$. Conversely, if $Z_mpartof Z_n$, then by (7) must also $mpartof Z_n$, because by (99) $mpartof Z_m$, which was to be proved.

mypara{101.} Theorem. The conditions for the cases $lambda$, $mu$, $nu$ in (90) may also be put in the following form: begin{eqnarray*} lambda. & quad m=n,quad n=m,quad Z_m=Z_n\ mu. & quad m<n,quad n>m,quad Z_{m'}partof Z_n\ nu. & quad m>n,quad n<m,quad Z_{n'}partof Z_m end{eqnarray*}

The proof follows immediately from (90) if we observe that by (100) the conditions $n_0partof m_0$ and $Z_mpartof Z_n$ are equivalent.

mypara{102.} Theorem. $Z_1=1$.

Proof. For by (99) the base-number 1 is contained in $Z_1$, while by (78) every number different from $1$ is contained in $1'_0$, hence by (98) not in $Z_1$, which was to be proved.

mypara{103.} Theorem. By (98) $N=mathfrak{M}(Z_n, n'_0)$.

mypara{104.} Theorem. $n=mathfrak{G}(Z_n, n_0)$, i.~e., $n$ is the only common element of the system $Z_n$ and $n_0$.

Proof. From (99) and (74) it follows that $n$ is % [File: 082.png] contained in $Z_n$ and $n_0$; but every element of the chain $n_0$, different from $n$ by (77) is contained in $n'_0$, and hence by (98) not in $Z_n$, which was to be proved.

mypara{105.} Theorem. By (91), (98) the number $n'$ is not contained in $Z_n$.

mypara{106.} Theorem. If $m<n$, then is $Z_m$ proper part of $Z_n$ and conversely.

Proof. If $m<n$, then by (100) $Z_m partof Z_n$, and since the number $n$, by (99) contained in $Z_n$, can by (98) not be contained in $Z_m$ because $n>m$, therefore $Z_m$ is proper part of $Z_n$. Conversely if $Z_m$ is proper part of $Z_n$ then by (100) $mleqq n$, and since $m$ cannot be $=n$, because otherwise $Z_m=Z_n$, we must have $m<n$, which was to be proved.

mypara{107.} Theorem. $Z_n$ is proper part of $Z_{n'}$.

The proof follows from (106), because by (91) $n<n'$.

mypara{108.} Theorem. $Z_{n'}= mathfrak{M}(Z_n,n')$.

Proof. For every number contained in $Z_{n'}$, by (98) is $leqq n'$, hence either $=n'$ or $<n'$, and therefore by (98) element of $Z_n$. Therefore certainly $Z_{n'} partof mathfrak{M}(Z_n,n')$. Since conversely by (107) $Z_n partof Z_{n'}$ and by (99) $n' partof Z_{n'}$, then by (10) we have [ mathfrak{M}(Z_n,n') partof Z_{n'}, ] whence our theorem follows by (5).

mypara{109.} Theorem. The transform $Z'_n$ of the system $Z_n$ is proper part of the system $Z_{n'}$.

Proof. For every number contained in $Z'_n$ is the transform $m'$ of a number $m$ contained in $Z_n$, and since % [File: 083.png] $m leqq n$, and hence by (94) $m' leqq n'$, we have by (98) $Z'_n partof Z_{n'}$. Since further the number 1 by (99) is contained in $Z_{n'}$, but by (71) is not contained in the transform $Z'_n$, then is $Z'_n$ proper part of $Z_{n'}$, which was to be proved.

mypara{110.} Theorem. $Z_{n'} = mathfrak{M}(1, Z'_n)$.

Proof. Every number of the system $Z_{n'}$ different from 1 by (78) is the transform $m'$ of a number $m$ and this must be $leqq n$, and hence by (98) contained in $Z_n$ (because otherwise $m>n$, 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) $<g$, which is contrary to what precedes; hence $g$ is contained in $E$, which was to be proved.

mypara{115.} Definition. If $l<m$ and $m<n$ we say the number $m$ textit{lies between} $l$ and $n$ (also between $n$ and $l$).

mypara{116.} Theorem. There exists no number lying between $n$ and $n'$.

Proof. For as soon as $m < n'$, and hence by (93) % [File: 085.png] $mleqq n$, then by (90) we cannot have $n<m$, which was to be proved.

mypara{117.} Theorem. If $t$ is a number in $T$, but not the least (96), then there exists in $T$ one and only one textit{next less} number $s$, i.~e., a number $s$ such that $s<t$, and that there exists in $T$ no number lying between $s$ and $t$. Similarly, if $t$ is not the greatest number in $T$ (111) there always exists in $T$ one and only one textit{next greater} number $u$, i.~e., a number $u$ such that $t<u$, and that there exists in $T$ no number lying between $t$ and $u$. At the same time in $T$ $t$ is next greater than $s$ and next less than $u$.

Proof. If $t$ is not the least number in $T$, then let $E$ be the system of all those numbers of $T$ that are $<t$; then by (98) $E partof Z_t$, and hence by (114) there exists in $E$ a greatest number $s$ obviously possessing the properties stated in the theorem, and also it is the only such number. If further $t$ is not the greatest number in $T$, then by (96) there certainly exists among all the numbers of $T$, that are $>t$, 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 $m<n$; then by (106) $Z_m$ is proper part of $Z_n$, and since by (119) $Z_n$ is finite, then by (64) $Z_m$ and $Z_n$ cannot be similar, which was to be proved.

mypara{121.} Theorem. Every part $E$ of the number-series $N$, which possesses a greatest number (111), is finite.

The proof follows from (113), (119), (68).

mypara{122.} Theorem. Every part $U$ of the number-series $N$, which possesses no greatest number, is simply infinite (71).

Proof. If $u$ is any number in $U$, there exists in $U$ by (117) one and only one next greater number than $u$, which we will denote by $psi(u)$ and regard as transform of $u$. The thus perfectly determined transformation $psi$ of the system $U$ has obviously the property %[ % alpha.quad psi(U) partof U, %] \ nullhfill $alpha.quad psi(U) partof U$, hfillnull\ i.~e., $U$ transformed in itself by $psi$. If further $u$, $v$ % [File: 087.png] are different numbers in $U$, then by symmetry we may by (90) assume that $u<v$; thus by (117) it follows from the definition of $psi$ that $psi(u) leqq v$ and $v < psi(v)$, and hence by (95) $psi(u) < psi(v)$; therefore by (90) the transforms $psi(u)$, $psi(v)$ are different, i.~e., %begin{quote} %[**F2: Odd bits. {quote} and/or {center} puts whitespace above and below the lines which isn't in the image.] \ nullhfill $delta$. the transformation $psi$ is similar. hfillnull\ %end{quote} Further, if $u_1$ denotes the least number (96) of the system $U$, then every number $u$ contained in $U$ is $geqq u_1$, and since generally $u < psi(u)$, then by (95) $u_1<psi(u)$, and therefore by (90) $u_1$ is different from $psi(u)$, i.~e., \ nullhfill %begin{quote} $gamma$. the element $u_1$ of $U$ is not contained in $psi(u)$. %end{quote} hfillnull\ Therefore $psi(U)$ is proper part of $U$ and hence by (64) $U$ is an infinite system. If then in agreement with (44) we denote by $psi_0(V)$, when $V$ is any part of $U$, the chain of $V$ corresponding to the transformation $psi$, we wish to show finally that \ nullhfill %begin{quote} $beta$. $U=psi_0(u_1)$. %end{quote} hfillnull\ In fact, since every such chain $psi_0(V)$ by reason of its definition (44) is a part of the system $U$ transformed in itself by $psi$, then evidently is $psi_0(u_1) partof U$; conversely it is first of all obvious from (45) that the element $u_1$ contained in $U$ is certainly contained in $psi_0(u_1)$; but if we assume that there exist elements of $U$, that are not contained in $psi_0(u_1)$, then must there be among them by (96) a least number $w$, and since by what precedes this is different from the least number $u_1$ of the system $U$, then by (117) must there exist in $U$ also a number $v$ which is next less than $w$, whence it % [File: 088.png] follows at once that $w = phi(v)$; since therefore $v<w$, then must $v$ by reason of the definition of $w$ certainly be contained in $psi_0(u_1)$; but from this by (55) it follows that also $psi(v)$, and hence $w$ must be contained in $psi_0(u_1)$, and since this is contrary to the definition of $w$, our foregoing hypothesis is inadmissible; therefore $U partof psi_0(u_1)$ and hence also $U=psi_0(u_1)$ as stated. From $alpha$, $beta$, $gamma$, $delta$ it then follows by (71) that $U$ is a simply infinite system set in order by $psi$, which was to be proved.

mypara{123.} Theorem. In consequence of (121), (122) any part $T$ of the number-series $N$ is finite or simply infinite, according as a greatest number exists or does not exist in $T$.

%Observed to be close to bottom of page so force skip newpage %IX.

mysect{VII}{IX}{Definition of a Transformation of the Number-Series by Induction}

mypara{124.} In what follows we denote numbers by small Italics and retain throughout all symbols of the previous sections VI.to VIII., while $Omega$ designates an arbitrary system whose elements are not necessarily contained in $N$.

mypara{125.} Theorem. If there is given an arbitrary (similar or dissimilar) transformation $theta$ of a system $Omega$ in itself, and besides a determinate element $omega$ in $Omega$, then to every number $n$ corresponds one transformation $psi_n$ and one only of the associated number-system $Z_n$ explained in (98), which satisfies the conditions:footnote

{For clearness here and in the following theorem (126) I have especially

mentioned condition I., although properly it is a consequence of II.and III.}

% [File: 089.png] parbegin{tabular}[b]{@{}r@{}l@{}}

&I@. $psi_n{Z_n} partof Omega$ \

I&I@. $psi_n(1)=omega$ \

II&I@. $psi_n(t')=thetapsi_n(t)$, end{tabular} if $t<n$, where the symbol $thetapsi_n$ has the meaning given in (25).

Proof by complete induction (80).

$rho$. label{psi1}The theorem is true for $n = 1$. In this case indeed by (102) the system $Z_{n}$ consists of the single number 1, and the transformation $psi_1$ %[**Corrected from comma, noted at end.] is therefore completely defined by II alone so that I is fulfilled while III drops out entirely.

$sigma$. If the theorem is true for a number $n$ then we show that it is also true for the following number $p=n'$, and we begin by proving that there can be only a single corresponding transformation $psi_p$ of the system $Z_p$. In fact, if a transformation $psi_p$ satisfies the conditions parbegin{tabular}[b]{@{}r@{}l@{}}

&I$'$. $psi_p(Z_p) partof Omega$ \

I&I$'$. $psi_p(1)=omega$\

II&I$'$. $psi_p(m')=thetapsi_p(m)$, end{tabular} when $m<p$, then there is also contained in it by (21), because $Z_n partof Z_p$ (107) a transformation of $Z_{n}$ which obviously satisfies the same conditions I, II, III as $psi_n$, and therefore coincides throughout with $psi_n$; for all numbers contained in $Z_{n}$, and hence (98) for all numbers $m$ which are $<p$, i.~e., $leqq n$, must therefore [ psi_p(m)=psi_n(m) tag{$m$} ] whence there follows, as a special case, [ psi_p(n)=psi_n(n); tag{$n$} ] since further by (105), (108) $p$ is the only number of % [File: 090.png] the system $Z_p$ not contained in $Z_n$, and since by III$'$ and $(n)$ we must also have [ psi_p(p) = thetapsi_n(n) tag{$p$} ] there follows the correctness of our foregoing statement that there can be only one transformation $psi_p$ of the system $Z_p$ satisfying the conditions I$'$, II$'$, III$'$, because by the conditions $(m)$ and $(p)$ just derived $psi_p$ is completely reduced to $psi_n$. We have next to show conversely that this transformation $psi_p$ of the system $Z_p$ completely determined by $(m)$ and $(p)$ actually satisfies the conditions I$'$, II$'$, III$'$. Obviously I$'$ follows from $(m)$ and $(p)$ with reference to I, and because $theta(Omega) partof Omega$. Similarly II$'$ follows from $(m)$ and II, since by (99) the number 1 is contained in $Z_n$. The correctness of III$'$ follows first for those numbers $m$ which are $<n$ from $(m)$ and III, and for the single number $m=n$ yet remaining it results from $(p)$ and $(n)$. Thus it is completely established that from the validity of our theorem for the number $n$ always follows its validity for the following number $p$, which was to be proved.

mypara{126.} Theorem of the definition by induction. If there is given an arbitrary (similar or dissimilar) transformation $theta$ of a system $Omega$ in itself, and besides a determinate element $omega$ in $Omega$, then there exists one and only one transformation $psi$ of the number-series $N$, which satisfies the conditions parbegin{tabular}[b]{@{}r@{}l@{}}

&I@. $psi(N) partof Omega$\

I&I@. $psi(1)=omega$\ % [File: 091.png] II&I@. $psi(n') = thetapsi(n)$, end{tabular} where $n$ represents every number.

Proof. Since, if there actually exists such a transformation $psi$, there is contained in it by (21) a transformation $psi_n$, of the system $Z_n$, which satisfies the conditions I, II, III stated in (125), then because there exists one and only one such transformation $psi_n$ must necessarily [ psi(n) =psi_n(n). tag{$n$} ] Since thus $psi$ is completely determined it follows also that there can exist only one such transformation $psi$ (see the closing remark in (130)). That conversely the transformation $psi$ determined by $(n)$ also satisfies our conditions I, II, III, follows easily from $(n)$ with reference to the properties I, II and $(p)$ shown in (125), which was to be proved.

mypara{127.} Theorem. Under the hypotheses made in the foregoing theorem, [ psi(T') = thetapsi(T), ] where $T$ denotes any part of the number-series $N$.

Proof. For if $t$ denotes every number of the system $T$, then $psi(T')$ consists of all elements $psi(t')$, and $thetapsi(T)$ of all elements $thetapsi(t)$; hence our theorem follows because by III in (126) $psi(t') = thetapsi(t)$.

mypara{128.} Theorem. If we maintain the same hypotheses and denote by $theta_{0}$ the chains (44) which correspond to the transformation $theta$ of the system $Omega$ in itself, then is [ psi(N) = theta_0(omega). ]

% [File: 092.png]

Proof. We show first by complete induction (80) that [ psi(N) partof theta_0(omega), ] i.~e., that every transform $psi(n)$ is also element of $theta_0(omega)$. In fact,

$rho$. this theorem is true for $n = 1$, because by (126, II) $psi(1)=omega$, and because by (45) $omega partof theta_0(omega)$.

$sigma$. If the theorem is true for a number $n$, and hence $psi(n) partof theta_0(omega)$, then by (55) also $thetabigl(psi(n)bigr) partof theta_0(omega)$, i.~e., by (126, III) $psi(n') partof theta_0(omega)$, hence the theorem is true for the following number $n'$, which was to be proved.

In order further to show that every element $nu$ of the chain $theta_0(omega)$ is contained in $psi(N)$, therefore that [ theta_0(omega) partof psi(N) ] we likewise apply complete induction, i.~e., theorem (59) transferred to $Omega$ and the transformation $theta$. In fact,

$rho$. the element $omega=psi(1)$, and hence is contained in $psi(N)$.

$sigma$. If $nu$ is a common element of the chain $theta_0(omega)$ and the system $psi(N)$, then $nu=psi(n)$, where $n$ denotes a number, and by (126, III) we get $theta(nu) = thetapsi(n) = psi(n')$, and therefore $theta(nu)$ is contained in $psi(N)$, which was to be proved.

From the theorems just established, $psi(N) partof theta_0(omega)$ and $theta_0(omega) partof psi(N)$, we get by (5) $psi(N) = theta_0(omega)$, which was to be proved.

mypara{129.} Theorem. Under the same hypotheses we have generally: % [File: 093.png] [ psi(n_0) = theta_0bigl(psi(n)bigr). ]

Proof by complete induction (80). For

$rho$. By (128) the theorem holds for $n = 1$, since $1_0=N$ and $psi(1)=omega$.

$sigma$. If the theorem is true for a number $n$, then [

thetabigl(psi(n_0)bigr)

= thetabigl(theta_0bigl(psi(n)bigr)bigr); ] since by (127), (75) [ thetabigl(psi(n_0)bigr) = psi(n'_0), ] and by (57), (126, III) [

thetabigl(theta_0bigl(psi(n)bigr)bigr)

= theta_0bigl(thetabigl(psi(n)bigr)bigr) = theta_0bigl(psi(n')bigr), ] rlap{we get} hfill$ psi(n'_0) = theta_0bigl(psi(n')bigr)$, hfillnull\ i.~e., the theorem is true for the number $n'$ following $n$, which was to be proved.

mypara{130.} Remark. Before we pass to the most important applications of the theorem of definition by induction proved in (126), (sections X--XIV), it is worth while to call attention to a circumstance by which it is essentially distinguished from the theorem of demonstration by induction proved in (80) or rather in (59), (60), however close may seem the relation between the former and the latter. For while the theorem (59) is true quite generally for every chain $A_0$ where $A$ is any part of a system $S$ transformed in itself by any transformation $phi$ (IV), the case is quite different with the theorem (126), which declares only the existence of a consistent (or one-to-one) transformation $psi$ of the simply infinite system $1_0$. If in the latter theorem (still maintaining the hypotheses regarding $Omega$ and $theta$) we replace the number-series $1_0$ by an arbitrary % [File: 094.png] chain $A_0$ out of such a system $S$, and define a transformation $psi$ of $A_0$ in $Omega$ in a manner analogous to that in (126, II, III) by assuming that

$rho$. to every element $a$ of $A$ there is to correspond a determinate element $psi(a)$ selected from $Omega$, and

$sigma$. for every element $n$ contained in $A_0$ and its transform $n'=phi(n)$, the condition $psi(n')=theta psi(n)$ is to hold, then would the case very frequently occur that such a transformation $psi$ does not exist, since these conditions $rho$, $sigma$ may prove incompatible, even though the freedom of choice contained in $rho$ be restricted at the outset to conform to the condition $sigma$. An example will be sufficient to convince one of this. If the system $S$ consisting of the different elements $a$ and $b$ is so transformed in itself by $phi$ that $a'=b$, $b'=a$, then obviously $a_0=b_0=S$; suppose further the system $Omega$ consisting of the different elements $alpha$, $beta$, and $gamma$ be so transformed in itself by $theta$ that $theta(alpha)=beta$, $theta(beta)=gamma$, $theta(gamma)=alpha$; if we now demand a transformation $psi$ of $a_0$ in $Omega$ such that $psi(a)=alpha$, and that besides for every element $n$ contained in $a_0$ always $psi(n')=thetapsi(n)$, we meet a contradiction; since for $n=a$, we get $psi(b)=theta(alpha)=beta$, and hence for $n=b$, we must have $psi(a)=theta(beta)=gamma$, while we had assumed $psi(a)=alpha$.

But if there exists a transformation $psi$ of $A_0$ in $Omega$, which satisfies the foregoing conditions $rho$, $sigma$ without contradiction, then from (60) it follows easily that it is completely determined; for if the transformation $chi$ satisfies the same conditions, then we have, generally, % [File: 095.png] $chi(n) = psi(n)$, since by $rho$ this theorem is true for all elements $n=a$ contained in $A$, and since if it is true for an element $n$ of $A_0$ it must by $sigma$ be true also for its transform $n'$.

mypara{131.} In order to bring out clearly the import of our theorem (126), we will here insert a consideration which is useful for other investigations also, e.~g., for the so-called group-theory.

We consider a system $Omega$, whose elements allow a certain combination such that from an element $nu$ by the effect of an element $omega$, there always results again a determinate element of the same system $Omega$, which may be denoted by $omegacenterdot nu$ or $omeganu$, and in general is to be distinguished from $nuomega$. We can also consider this in such a way that to every determinate element $omega$, there corresponds a determinate transformation of the system $Omega$ in itself (to be denoted by $dot{omega}$), in so far as every element $nu$ furnishes the determinate transform $dot{omega}(nu)= omega nu$. If to this system $Omega$ and its element $omega$ we apply theorem (126), designating by $dot{omega}$ the transformation there denoted by $theta$, then there corresponds to every number $n$ a determinate element $psi(n)$ contained in $Omega$, which may now be denoted by the symbol $omega^n$ and sometimes called the $n$th power of $omega$; this notion is completely defined by the conditions imposed upon it begin{align*} text{II@. }& omega^1 = omega\ text{III@. }& omega^{n'}! = omega, omega^n, end{align*} and its existence is established by the proof of theorem (126).

% [File: 096.png]

If the foregoing combination of the elements is further so qualified that for arbitrary elements $mu$, $nu$, $omega$, we always have $omega(nu,mu)$ = $omega,nu(mu)$, then are true also the theorems [

omega^{n'}! = omega^n, omega, quad omega^m, omega^n = omega^n, omega^m,

] whose proofs can easily be effected by complete induction and may be left to the reader.

The foregoing general consideration may be immediately applied to the following example. If $S$ is a system of arbitrary elements, and $Omega$ the associated system whose elements are all the transformations $nu$ of $S$ in itself (36), then by (25) can these elements be continually compounded, since $nu(S) partof S$, and the transformation $omeganu$ compounded out of such transformations $nu$ and $omega$ is itself again an element of $Omega$. Then are also all elements $omega^n$ transformations of $S$ in itself, and we say they arise by repetition of the transformation $omega$. We will now call attention to a simple connection existing between this notion and the notion of the chain $omega_0(A)$ defined in (44), where $A$ again denotes any part of $S$. If for brevity we denote by $A_n$ the transform $omega^n(A)$ produced by the transformation $omega^n$, then from III and (25) it follows that $omega(A_n)=A_{n'}$. Hence it is easily shown by complete induction (80) that all these systems $A_n$ are parts of the chain $omega_0(A)$; for

$rho$. by (50) this statement is true for $n = 1$, and

$sigma$. if it is true for a number $n$, then from (55) and from $A_{n'}=omega(A_n)$ it follows that it is also true for the following number $n'$, which was to be proved. Since % [File: 097.png] further by (45) $A partof omega_0(A)$, then from (10) it results that the system $K$ compounded out of $A$ and all transforms $A_n$ is part of $omega_0(A)$. Conversely, since by (23) $omega(K)$ is compounded out of $omega(A)=A_1$ and all systems $omega(A_n)=A_{n'}$, therefore by (78) out of all systems $A_n$, which by (9) are parts of $K$, then by (10) is $omega(K) partof K$, i.~e., $K$ is a chain (37), and since by (9) $A partof K$, then by (47) it follows also that that $omega_0(A) partof K$. Therefore $omega_0(A)=K$, i.~e., the following theorem holds: If $omega$ is a transformation of a system $S$ in itself, and $A$ any part of $S$, then is the chain of $A$ corresponding to the transformation $omega$ compounded out of $A$ and all the transforms $omega^n(A)$ resulting from repetitions of $omega$. We advise the reader with this conception of a chain to return to the earlier theorems (57), (58).

%X.

mysect{VIII}{X}{The Class of Simply Infinite Systems}

mypara{132.} Theorem. All simply infinite systems are similar to the number-series $N$ and consequently by (33) also to one another.

Proof. Let the simply infinite system $Omega$ be set in order (71) by the transformation $theta$, and let $omega$ be the base-element of $Omega$ thus resulting; if we again denote by $theta_0$ the chains corresponding to the transformation $theta$ (44), then by (71) is the following true:

$alpha$. $theta(Omega) partof Omega$.

$beta$. $Omega = theta_0(omega)$.

% [File: 098.png]

$gamma$. $omega$ is not contained in $theta(Omega)$.

$delta$. The transformation $theta$ is similar.

noindent If then $psi$ denotes the transformation of the number-series $N$ defined in (126), then from $beta$ and (128) we get first [ psi(N)=Omega, ] and hence we have only yet to show that $psi$ is a similar transformation, i.~e., (26) that to different numbers $m$, $n$ correspond different transforms $psi(m)$, $psi(n)$. On account of the symmetry we may by (90) assume that $m>n$, 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 $thetapsi(n_0)$. 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 $thetapsi(n'_0)$, 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') = thetapsi(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 $theta=psiphioverline{psi}$, $phi=overline{psi}thetapsi$. 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 conditionsfootnote{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 $psi(1)=m'$, $psi(n')=phipsi(n)$ 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 $bigl((l+m)+nbigr)'=bigl(l+(m+n)bigr)'$, 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 $m<a$, then by (143) respectively $m+n> a+n$ or $m+n<a+n$, therefore by (90) we surely cannot have $m+n=a+n$ which was to be proved.

mypara{146.} Theorem. If $l>n$, 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 $mcenterdot n$ 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@. }& mcenterdot 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. $1centerdot n = n$.

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 $1centerdot n+1=n+1$, i.~e., by (147, III), (135, II) $1centerdot n'=n'$, 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 $psi(n')=thetapsi(n)$, 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'}=acenterdot a^n =rlap{$a^ncenterdot a$}. end{align*}

mypara{156.} Theorem. $a^{m+n}=a^mcenterdot a^n$.

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^mcenterdot 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) $(a^mcenterdot a^n)a=a^m(a^ncenterdot a)=a^mcenterdot a^{n'}$; hence $a^{m+n'}=a^mcenterdot a^{n'}$, 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)^ncenterdot a^m=a^{mn}centerdot a^m ] but by (155, III) $(a^m)^ncenterdot a^m=(a^m)^{n'}$, 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. $(ab)^n=a^ncenterdot b^n$

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 $(ab)^ncenterdot a= a(a^ncenterdot b^n)=(acenterdot a^n)b^n=a^{n'}centerdot b^n$, and thus $bigl( (ab)^ncenterdot a bigr)b= (a^{n'}centerdot b^n)b$; but by (153), (155, III) $bigl( (ab)^ncenterdot a bigr)b= (ab)^ncenterdot (ab)=(ab)^{n'}$, and likewise [

(a^{n'}centerdot b^n)b = a^{n'}centerdot (b^ncenterdot 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 $chi(N) partof Sigma$. If further $m$, $n$ are different numbers we may by reason of symmetry according to (90) suppose $m<n$; then by the foregoing $chi(m)=psi_m(m)=psi_n(m)$, and $chi(n)=psi_n(n)$; but since $psi_n$ was a similar transformation of $Z_n$ in $Sigma$, and $m$, $n$ are different elements of $Z_n$, then is $psi_n(m)$ different from $psi_n(n)$, hence also $chi(m)$ different from $chi(n)$, i.~e., $chi$ is a similar transformation of $N$. Since further $N$ is an infinite system (71), the same thing % [File: 114.png] is true by (67) of the system $chi(N)$ similar to it and by (68), because $chi(N)$ is part of $Sigma$, also of $Sigma$, which was to be proved.

mypara{160.} Theorem. A system $Sigma$ is finite or infinite, according as there does or does not exist a system $Z_n$ similar to it.

Proof. If $Sigma$ is finite, then by (159) there exist systems $Z_n$ which are not similarly transformable in $Sigma$; since by (102) the system $Z_1$ consists of the single number 1, and hence is similarly transformable in every system, then must the least number $k$ (96) to which a system $Z_k$ not similarly transformable in $Sigma$ corresponds be different from 1 and hence by (78) $=n'$, and since $n<n'$ (91) there exists a similar transformation $psi$ of $Z_n$ in $Sigma$; if then $psi(z_n)$ were only a proper part of $Sigma$, i.~e., if there existed an element $alpha$ in $Sigma$ not contained in $psi(Z_n)$, then since $Z_{n'}=mathfrak{M}(Z_n, n')$ (108) we could extend this transformation $psi$ to a similar transformation $psi$ of $Z_{n'}$ in $Sigma$ by putting $psi(n')=alpha$ while by our hypothesis $Z_{n'}$ is not similarly transformable in $Sigma$. Hence $psi(Z_n) = Sigma$, i.~e., $Z_n$ and $Sigma$ are similar systems. Conversely, if a system $Sigma$ is similar to a system $Z_n$, then by (119), (67) $Sigma$ is finite, which was to be proved.

mypara{161.} Definition. If $Sigma$ is a finite system, then by (160) there exists one and by (120), (33) only one single number $n$ to which a system $Z_n$ similar to the system $Sigma$ corresponds; this number $n$ is called the textit{number} [textit{Anzahl}] of the elements contained in $Sigma$ (or % [File: 115.png] also the textit{degree} of the system $Sigma$) and we say $Sigma$ consists of or is a system of $n$ elements, or the number $n$ shows textit{how many} elements are contained in $Sigma$.footnote

{For clearness and simplicity in what follows we restrict the notion of the number throughout to finite systems; if then we speak of a number of certain things, it is always understood that the system whose elements these things are is a finite system.}

If numbers are used to express accurately this determinate property of finite systems they are called textit{cardinal numbers}. As soon as a determinate similar transformation $psi$ of the system $Z_n$ is chosen by reason of which $psi(Z_n)=Z$, then to every number $m$ contained in $Z_n$ (i.~e., every number $m$ which is $leqq n$) there corresponds a determinate element $psi(m)$ of the system $Sigma$, and conversely by (26) to every element of $Sigma$ by the inverse transformation $overline{psi}$ there corresponds a determinate number $m$ in $Z_n$. Very often we denote all elements of $Sigma$ by a single letter, e.~g., $alpha$, to which we append the distinguishing numbers $m$ as indices so that $psi(m)$ is denoted by $alpha_m$. We say also that these elements are textit{counted and set in order} by $psi$ in determinate manner, and call $alpha_m$ the $m$th element of $Sigma$; if $m<n$ then $alpha_{m'}$ is called the element textit{following} $alpha_m$, and $alpha_n$ is called the textit{last} element. In this counting of the elements therefore the numbers $m$ appear again as ordinal numbers (73).

mypara{162.} Theorem. All systems similar to a finite system possess the same number of elements.

The proof follows immediately from (33), (161).

mypara{163.} Theorem. The number of numbers contained in $Z_n$, i.~e., of those numbers which are $leqq n$, is $n$.

% [File: 116.png]

Proof. For by (32) $Z_n$ is similar to itself.

mypara{164.} Theorem. If a system consists of a single element, then is the number of its elements $=1$, and conversely.

The proof follows immediately from (2), (26), (32), (102), (161).

mypara{165.} Theorem. If $T$ is proper part of a finite system $Sigma$, then is the number of the elements of $T$ less than that of the elements of $Sigma$.

Proof. By (68) $T$ is a finite system, therefore similar to a system $Z_m$, where $m$ denotes the number of the elements of $T$; if further $n$ is the number of elements of $Sigma$, therefore $Sigma$ similar to $Z_n$, then by (35) $T$ is similar to a proper part $E$ of $Z_n$ and by (33) also $Z_m$ and $E$ are similar to each other; if then we were to have $n leqq m$, hence $Z_n partof Z_m$, by (7) $E$ would also be proper part of $Z_m$, and consequently $Z_m$ an infinite system, which contradicts theorem (119); hence by (90), $m<n$, which was to be proved.

mypara{166.} Theorem. If $Gamma=mathfrak{M}(B,gamma)$, where $B$ denotes a system of $n$ elements, and $gamma$ an element of $Gamma$ not contained in $B$, then $Gamma$ consists of $n'$ elements.

Proof. For if $B=psi(Z_n)$, where $psi$ denotes a similar transformation of $Z_n$, then by (105), (108) it may be extended to a similar transformation $psi$ of $Z_{n'}$, by putting $psi(n')=gamma$, and we get $psi(Z_{n'})=Gamma$, which was to be proved.

mypara{167.} Theorem. If $gamma$ is an element of a system $Gamma$ % [File: 117.png] consisting of $n'$ elements, then is $n$ the number of all other elements of $Gamma$.

Proof. For if $B$ denotes the aggregate of all elements in $Gamma$ different from $gamma$, then is $Gamma=mathfrak{M}(B,gamma)$; if then $b$ is the number of elements of the finite system $B$, by the foregoing theorem $b'$ is the number of elements of $Gamma$, therefore $=n'$, whence by (26) we get $b=n$, which was to be proved.

mypara{168.} Theorem. If $A$ consists of $m$ elements, and $B$ of $n$ elements, and $A$ and $B$ have no common element, then $mathfrak{M}(A,B)$ consists of $m+n$ elements.

Proof by complete induction (80). For

$rho$. by (166), (164), (135, II) 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 $n'$. In fact, if $Gamma$ is a system of $n'$ elements, then by (167) we can put $Gamma=mathfrak{M}(B,gamma)$ where $gamma$ denotes an element and $B$ the system of the $n$ other elements of $Gamma$. If then $A$ is a system of $m$ elements each of which is not contained in $Gamma$, therefore also not contained in $B$, and we put $mathfrak{M}(A,B)=Sigma$, by our hypothesis $m+n$ the number of elements of $Sigma$, and since $gamma$ is not contained in $Sigma$, then by (166) the number of elements contained in $mathfrak{M}(Sigma,gamma)=(m+n')$, therefore by (135, III) $=m+n'$; but since by (15) obviously $mathfrak{M}(Sigma,gamma)=mathfrak{M}(A,B,gamma)= mathfrak{M}(A,Gamma)$, then is $m+n'$ the number of the elements of $mathfrak{M}(A,Gamma)$, which was to be proved.

mypara{169.} Theorem. If $A$, $B$ are finite systems of $m$, $n$ % [File: 118.png] elements respectively, then is $mathfrak{M}(A,B)$ a finite system and the number of its elements is $leqq m+n$.

Proof. If $B partof A$, then $mathfrak{M}(A,B)=A$ and the number $m$ of the elements of this system is by (142) $<m+n$, as was stated. But if $B$ is not part of $A$, and $T$ is the system of all those elements of $B$ that are not contained in $A$, then by (165) is their number $p leqq n$, and since obviously [ mathfrak{M}(A,B)=mathfrak{M}(A,T), ] then by (143) is the number $m+p$ of the elements of this system $leqq m+n$, which was to be proved.

mypara{170.} Theorem. Every system compounded out of a number $n$ of finite systems is finite.

Proof by complete induction (80). For

$rho$. by (8) the theorem is self-evident for $n = 1$.

$sigma$. If the theorem is true for a number $n$, and if $Sigma$ is compounded out of $n'$ finite systems, then let $A$ be one of these systems and $B$ the system compounded out of all the rest; since their number by (167) $=n$, then by our hypothesis $B$ is a finite system. Since obviously $Sigma=mathfrak{M}(A,B)$, it follows from this and from (169) that $Sigma$ is also a finite system, which was to be proved.

mypara{171.} Theorem. If $psi$ is a dissimilar transformation of a finite system $Sigma$ of $n$ elements, then is the number of elements of the transform $psi(Sigma)$ less than $n$.

Proof. If we select from all those elements of $Sigma$ that possess one and the same transform, always one and only one at pleasure, then is the system $T$ of all % [File: 119.png] these selected elements obviously a proper part of $Sigma$, because $psi$ is a dissimilar transformation of $Sigma$ (26). At the same time it is clear that the transformation by (21) contained in $psi$ of this part $T$ is a similar transformation, and that $psi(T)=psi(Sigma)$; hence the system $psi(Sigma)$ is similar to the proper part $T$ of $Sigma$, and consequently our theorem follows by (162), (165).

mypara{172.} Final remark. Although it has just been shown that the number $m$ of the elements of $psi(Sigma)$ is less than the number $n$ of the elements of $Sigma$, yet in many cases we like to say that the number of elements of $psi(Sigma)=n$. The word number is then, of course, used in a different sense from that used hitherto (161); for if $alpha$ is an element of $Sigma$ and $a$ the number of all those elements of $Sigma$, that possess one and the same transform $psi(alpha)$ then is the latter as element of $psi(Sigma)$ frequently regarded still as representative of $a$ elements, which at least from their derivation may be considered as different from one another, and accordingly counted as $a$-fold element of $psi(Sigma)$. In this way we reach the notion, very useful in many cases, of systems in which every element is endowed with a certain frequency-number which indicates how often it is to be reckoned as element of the system. In the foregoing case, e.~g., we would say that $n$ is the number of the elements of $psi(Sigma)$ counted in this sense, while the number $m$ of the actually different elements of this system coincides with the number of the elements of $T$. Similar deviations from the original % [File: 120.png] meaning of a technical term which are simply extensions of the original notion, occur very frequently in mathematics; but it does not lie in the line of this memoir to go further into their discussion.

% [File: 121.png] %[Blank Page] % [File: 122.png]

begin{center}textsc{Typographical Errors corrected in Project Gutenberg edition}end{center}

p.~pageref{chain} ``$A' partof B$, $B' partof B$, $C' partof C$'' in original, amended to ``$A' partof A$'' etc@.

p.~pageref{psi1} ``the transformation $psi$, is therefore completely defined'' in original, amended to ``$psi_1$''.