Dedekind: Continuity and Irrational Numbers

Preface

My attention was first directed toward the considerations which form the subject of this pamphlet in the autumn of 1858. As professor in the Polytechnic School in Zürich I found myself for the first time obliged to lecture upon the elements of the differential calculus and felt more keenly than ever before the lack of a really scientific foundation for arithmetic. In discussing the notion of the approach of a variable magnitude to a fixed limiting value, and especially in proving the theorem that every magnitude which grows continually, but not beyond all limits, must certainly approach a limiting value, I had recourse to geometric evidences. Even now such resort to geometric intuition in a first presentation of the differential calculus, I regard as exceedingly useful, from the didactic standpoint, and indeed indispensable, if one does not wish to lose too much time. But that this form of introduction into the differential calculus can make no claim to being scientific, no one will deny. For myself this feeling of dissatisfaction was so overpowering that I made the fixed resolve to keep meditating on the question till I should find a purely arithmetic and perfectly rigorous foundation for the principles of infinitesimal analysis. The statement is so frequently made that the differential calculus deals with continuous magnitude, and yet an explanation of this continuity is nowhere given; even the most rigorous expositions of the differential calculus do not base their proofs upon continuity but, with more or less consciousness of the fact, they either appeal to geometric notions or those suggested by geometry, or depend upon theorems which are never established in a purely arithmetic manner. Among these, for example, belongs the above-mentioned theorem, and a more careful investigation convinced me that this theorem, or any one equivalent to it, can be regarded in some way as a sufficient basis for infinitesimal analysis. It then only remained to discover its true origin in the elements of arithmetic and thus at the same time to secure a real definition of the essence of continuity. I succeeded Nov. 24, 1858, and a few days afterward I communicated the results of my meditations to my dear friend Durège with whom I had a long and lively discussion. Later I explained these views of a scientific basis of arithmetic to a few of my pupils, and here in Braunschweig read a paper upon the subject before the scientific club of professors, but I could not make up my mind to its publication, because, in the first place, the presentation did not seem altogether simple, and further, the theory itself had little promise. Nevertheless I had already half determined to select this theme as subject for this occasion, when a few days ago, March 14, by the kindness of the author, the paper Die Elemente der Funktionenlehre by E. Heine (Crelle's Journal), Vol. 74) came into my hands and confirmed me in my decision. In the main I fully agree with the substance of this memoir, and indeed I could hardly do otherwise, but I will frankly acknowledge that my own presentation seems to me to be simpler in form and to bring out the vital point more clearly. While writing this preface (March 20, 1872), I am just in receipt of the interesting paper Ueber die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen, by G. Cantor Math. Annalen, Vol.~5), for which I owe the ingenious author my hearty thanks. As I find on a hasty perusal, the axiom given in Section II. of that paper, aside from the form of presentation, agrees with what I designate in Section III. as the essence of continuity. But what advantage will be gained by even a purely abstract definition of real numbers of a higher type, I am as yet unable to see, conceiving as I do of the domain of real numbers as complete in itself.

%I.

mysect{XVII}{I}{Properties of Rational Numbers}

%hspace{parindent} The development of the arithmetic of rational numbers is here presupposed, but still I think it worth while to call attention to certain important % [File: 010.png---Firefly Diva\sahobartglimpseofchaoshutcheson----- matters without discussion, so as to show at the outset the standpoint assumed in what follows. I regard the whole of arithmetic as a necessary, or at least natural, consequence of the simplest arithmetic act, that of counting, and counting itself as nothing else than the successive creation of the infinite series of positive integers in which each individual is defined by the one immediately preceding; the simplest act is the passing from an already-formed individual to the consecutive new one to be formed. The chain of these numbers forms in itself an exceedingly useful instrument for the human mind; it presents an inexhaustible wealth of remarkable laws obtained by the introduction of the four fundamental operations of arithmetic. Addition is the combination of any arbitrary repetitions of the above-mentioned simplest act into a single act; from it in a similar way arises multiplication. While the performance of these two operations is always possible, that of the inverse operations, subtraction and division, proves to be limited. Whatever the immediate occasion may have been, whatever comparisons or analogies with experience, or intuition, may have led thereto; it is certainly true that just this limitation in performing the indirect operations has in each case been the real motive for a new creative act; thus negative and fractional numbers have been created by the human mind; and in the system of all rational numbers there has been gained an instrument of infinitely greater perfection. This system, % [File: 011.png---Firefly Diva\sahobartglimpseofchaoshutcheson----- which I shall denote by $R$, possesses first of all a completeness and self-containedness which I have designated in another placefootnote{textit{Vorlesungen über Zahlentheorie}, by P.~G.~Lejeune Dirichlet. 2d~ed. §159.} as characteristic of a textit{body of numbers} [Zahlkörper] and which consists in this that the four fundamental operations are always performable with any two individuals in $R$, i.~e., the result is always an individual of $R$, the single case of division by the number zero being excepted.

For our immediate purpose, however, another property of the system $R$ is still more important; it may be expressed by saying that the system $R$ forms a well-arranged domain of one dimension extending to infinity on two opposite sides. What is meant by this is sufficiently indicated by my use of expressions borrowed from geometric ideas; but just for this reason it will be necessary to bring out clearly the corresponding purely arithmetic properties in order to avoid even the appearance as if arithmetic were in need of ideas foreign to it.

To express that the symbols $a$ and $b$ represent one and the same rational number we put $a = b$ as well as $b = a$. The fact that two rational numbers $a$, $b$ are different appears in this that the difference $a - b$ has either a positive or negative value. In the former case $a$ is said to be textit{greater} than $b$, $b$ textit{less} than $a$; this is also indicated by the symbols $a > b$, $b < a$.footnote{Hence in what follows the so-called ``algebraic'' greater and less are understood unless the word ``absolute'' is added.} As in the latter case $b - a$ has a positive value it follows % [File: 012.png---Firefly Diva\sahobartglimpseofchaoshutcheson----- that $b > a$, $a < b$. In regard to these two ways in which two numbers may differ the following laws will hold:

textsc{i.} If $a > b$, and $b > c$, then $a > c$. Whenever $a$, $c$ are two different (or unequal) numbers, and $b$ is greater than the one and less than the other, we shall, without hesitation because of the suggestion of geometric ideas, express this briefly by saying: $b$ lies between the two numbers $a$, $c$.

textsc{ii.} If $a$, $c$ are two different numbers, there are infinitely many different numbers lying between $a$, $c$.

textsc{iii.} If $a$ is any definite number, then all numbers of the system $R$ fall into two classes, $A_1$ and $A_2$, each of which contains infinitely many individuals; the first class $A_1$ comprises all numbers $a_1$ that are $< a$, the second class $A_2$ comprises all numbers $a_2$ that are $> a$; the number $a$ itself may be assigned at pleasure to the first or second class, being respectively the greatest number of the first class or the least of the second. In every case the separation of the system $R$ into the two classes $A_1$, $A_2$ is such that every number of the first class $A_1$ is less than every number of the second class $A_2$.

%II.

mysect{XVI}{II}{Comparison of the Rational Numbers with the Points of a Straight Line}

The above-mentioned properties of rational numbers recall the corresponding relations of position of % [File: 013.png] the points of a straight line $L$. If the two opposite directions existing upon it are distinguished by ``right'' and ``left,'' and $p$, $q$ are two different points, then either $p$ lies to the right of $q$, and at the same time $q$ to the left of $p$, or conversely $q$ lies to the right of $p$ and at the same time $p$ to the left of $q$. A third case is impossible, if $p$, $q$ are actually different points. In regard to this difference in position the following laws hold:

textsc{i.} If $p$ lies to the right of $q$, and $q$ to the right of $r$, then $p$ lies to the right of $r$; and we say that $q$ lies between the points $p$ and $r$.

textsc{ii.} If $p$, $r$ are two different points, then there always exist infinitely many points that lie between $p$ and $r$.

textsc{iii.} If $p$ is a definite point in $L$, then all points in $L$ fall into two classes, $P_1$, $P_2$, each of which contains infinitely many individuals; the first class $P_1$ contains all the points $p_1$, that lie to the left of $p$, and the second class $P_2$ contains all the points $p_2$ that lie to the right of $p$; the point $p$ itself may be assigned at pleasure to the first or second class. In every case the separation of the straight line $L$ into the two classes or portions $P_1$, $P_2$, is of such a character that every point of the first class $P_1$ lies to the left of every point of the second class $P_2$.

This analogy between rational numbers and the points of a straight line, as is well known, becomes a real correspondence when we select upon the straight % [File: 014.png] line a definite origin or zero-point $o$ and a definite unit of length for the measurement of segments. With the aid of the latter to every rational number $a$ a corresponding length can be constructed and if we lay this off upon the straight line to the right or left of $o$ according as $a$ is positive or negative, we obtain a definite end-point $p$, which may be regarded as the point corresponding to the number $a$; to the rational number zero corresponds the point $o$. In this way to every rational number $a$, i.~e., to every individual in $R$, corresponds one and only one point $p$, i.~e., an individual in $L$. To the two numbers $a$, $b$ respectively correspond the two points, $p$, $q$, and if $a>b$, then $p$ lies to the right of $q$. To the laws textsc{i}, textsc{ii}, textsc{iii} of the previous Section correspond completely the laws textsc{i}, textsc{ii}, textsc{iii} of the present.

%III.

mysect{XV}{III}{Continuity of the Straight Line} label{EISIII}

Of the greatest importance, however, is the fact that in the straight line $L$ there are infinitely many points which correspond to no rational number. If the point $p$ corresponds to the rational number $a$, then, as is well known, the length $o,p$ is commensurable with the invariable unit of measure used in the construction, i.~e., there exists a third length, a so-called common measure, of which these two lengths are integral multiples. But the ancient Greeks already % [File: 015.png] knew and had demonstrated that there are lengths incommensurable with a given unit of length, e.~g., the diagonal of the square whose side is the unit of length. If we lay off such a length from the point $o$ upon the line we obtain an end-point which corresponds to no rational number. Since further it can be easily shown that there are infinitely many lengths which are incommensurable with the unit of length, we may affirm: The straight line $L$ is infinitely richer in point-individuals than the domain $R$ of rational numbers in number-individuals.

If now, as is our desire, we try to follow up arithmetically all phenomena in the straight line, the domain of rational numbers is insufficient and it becomes absolutely necessary that the instrument $R$ constructed by the creation of the rational numbers be essentially improved by the creation of new numbers such that the domain of numbers shall gain the same completeness, or as we may say at once, the same textit{continuity}, as the straight line.

The previous considerations are so familiar and well known to all that many will regard their repetition quite superfluous. Still I regarded this recapitulation as necessary to prepare properly for the main question. For, the way in which the irrational numbers are usually introduced is based directly upon the conception of extensive magnitudes---which itself is nowhere carefully defined---and explains number as the result of measuring such a magnitude by another % [File: 016.png] of the same kind.footnote{The apparent advantage of the generality of this definition of number disappears as soon as we consider complex numbers. According to my view, on the other hand, the notion of the ratio between two numbers of the same kind can be clearly developed only after the introduction of irrational numbers.} Instead of this I demand that arithmetic shall be developed out of itself.

That such comparisons with non-arithmetic notions have furnished the immediate occasion for the extension of the number-concept may, in a general way, be granted (though this was certainly not the case in the introduction of complex numbers); but this surely is no sufficient ground for introducing these foreign notions into arithmetic, the science of numbers. Just as negative and fractional rational numbers are formed by a new creation, and as the laws of operating with these numbers must and can be reduced to the laws of operating with positive integers, so we must endeavor completely to define irrational numbers by means of the rational numbers alone. The question only remains how to do this.

The above comparison of the domain $R$ of rational numbers with a straight line has led to the recognition of the existence of gaps, of a certain incompleteness or discontinuity of the former, while we ascribe to the straight line completeness, absence of gaps, or continuity. In what then does this continuity consist? Everything must depend on the answer to this question, and only through it shall we obtain a scientific basis for the investigation of textit{all} continuous domains. By vague remarks upon the unbroken connection in % [File: 017.png] the smallest parts obviously nothing is gained; the problem is to indicate a precise characteristic of continuity that can serve as the basis for valid deductions. For a long time I pondered over this in vain, but finally I found what I was seeking. This discovery will, perhaps, be differently estimated by different people; the majority may find its substance very commonplace. It consists of the following. In the preceding section attention was called to the fact that every point $p$ of the straight line produces a separation of the same into two portions such that every point of one portion lies to the left of every point of the other. I find the essence of continuity in the converse, i.~e., in the following principle:

``If all points of the straight line fall into two classes such that every point of the first class lies to the left of every point of the second class, then there exists one and only one point which produces this division of all points into two classes, this severing of the straight line into two portions.''

As already said I think I shall not err in assuming that every one will at once grant the truth of this statement; the majority of my readers will be very much disappointed in learning that by this commonplace remark the secret of continuity is to be revealed. To this I may say that I am glad if every one finds the above principle so obvious and so in harmony with his own ideas of a line; for I am utterly unable to adduce any proof of its correctness, nor has any % [File: 018.png] one the power. The assumption of this property of the line is nothing else than an axiom by which we attribute to the line its continuity, by which we find continuity in the line. If space has at all a real existence it is textit{not} necessary for it to be continuous; many of its properties would remain the same even were it discontinuous. And if we knew for certain that space was discontinuous there would be nothing to prevent us, in case we so desired, from filling up its gaps, in thought, and thus making it continuous; this filling up would consist in a creation of new point-individuals and would have to be effected in accordance with the above principle.

%IV.

mysect{XII}{IV}{Creation of Irrational Numbers} label{EISIV}

From the last remarks it is sufficiently obvious how the discontinuous domain $R$ of rational numbers may be rendered complete so as to form a continuous domain. In Section~I it was pointed out that every rational number $a$ effects a separation of the system $R$ into two classes such that every number $a_1$ of the first class $A_1$ is less than every number $a_2$ of the second class $A_2$; the number $a$ is either the greatest number of the class $A_1$ or the least number of the class $A_2$. If now any separation of the system $R$ into two classes $A_1$, $A_2$ is given which possesses only textit{this} characteristic property that every number $a_1$ in $A_1$ is less than every number $a_2$ in $A_2$, then for brevity we shall call % [File: 019.png] such a separation a textit{cut} [Schnitt] and designate it by $(A_1, A_2)$. We can then say that every rational number $a$ produces one cut or, strictly speaking, two cuts, which, however, we shall not look upon as essentially different; this cut possesses, textit{besides}, the property that either among the numbers of the first class there exists a greatest or among the numbers of the second class a least number. And conversely, if a cut possesses this property, then it is produced by this greatest or least rational number.

But it is easy to show that there exist infinitely many cuts not produced by rational numbers. The following example suggests itself most readily.

Let $D$ be a positive integer but not the square of an integer, then there exists a positive integer $lambda$ such that

[ lambda^2 < D <(lambda + 1)^2. ]

If we assign to the second class $A_2$, every positive rational number $a_2$ whose square is $> D$, to the first class $A_1$ all other rational numbers $a_1$, this separation forms a cut $(A_1, A_2)$, i.~e., every number $a_1$ is less than every number $a_2$. For if $a_1 = 0$, or is negative, then on that ground $a_1$ is less than any number $a_2$, because, by definition, this last is positive; if $a_1$ is positive, then is its square $leqq D$, and hence $a_1$ is less than any positive number $a_2$ whose square is $> D$.

But this cut is produced by no rational number. To demonstrate this it must be shown first of all that there exists no rational number whose square $= D$. % [File: 020.png] Although this is known from the first elements of the theory of numbers, still the following indirect proof may find place here. If there exist a rational number whose square $= D$, then there exist two positive integers $t$, $u$, that satisfy the equation [ t^2 - Du^2 = 0, ] and we may assume that $u$ is the textit{least} positive integer possessing the property that its square, by multiplication by $D$, may be converted into the square of an integer $t$. Since evidently [ lambda u < t < (lambda + 1) u, ] the number $u' = t - lambda u$ is a positive integer certainly textit{less} than $u$. If further we put [ t' = Du - lambda t, ] $t'$ is likewise a positive integer, and we have [ t'^2 - Du'^2 = (lambda^2 - D)(t^2 - Du^2) = 0, ] which is contrary to the assumption respecting $u$.

Hence the square of every rational number $x$ is either $< D$ or $> D$. From this it easily follows that there is neither in the class $A_1$ a greatest, nor in the class $A_2$ a least number. For if we put begin{align*} y &= frac{x(x^2 + 3D)}{3x^2 + D}, shortintertext {we have} y - x &= frac{2x(D - x^2)}{3x^2 + D} shortintertext {and} y^2 - D &= frac{(x^2 - D)^3}{(3x^2 + D)^2}. end{align*}

% [File: 021.png]

If in this we assume $x$ to be a positive number from the class $A_1$, then $x^2<D$, and hence $y>x$ and $y^2<D$. Therefore $y$ likewise belongs to the class $A_1$. But if we assume $x$ to be a number from the class $A_2$, then $x^2>D$, and hence $y<x$, $y>0$, and $y^2>D$. Therefore $y$ likewise belongs to the class $A_2$. This cut is therefore produced by no rational number.

In this property that not all cuts are produced by rational numbers consists the incompleteness or discontinuity of the domain $R$ of all rational numbers.

Whenever, then, we have to do with a cut $(A_1, A_2)$ produced by no rational number, we create a new, an textit{irrational} number $alpha$, which we regard as completely defined by this cut $(A_1, A_2)$; we shall say that the number $alpha$ corresponds to this cut, or that it produces this cut. From now on, therefore, to every definite cut there corresponds a definite rational or irrational number, and we regard two numbers as textit{different} or textit{unequal} always and only when they correspond to essentially different cuts.

In order to obtain a basis for the orderly arrangement of all textit{real}, i.~e., of all rational and irrational numbers we must investigate the relation between any two cuts $(A_1, A_2)$ and $(B_1, B_2)$ produced by any two numbers $alpha$ and $beta$. Obviously a cut $(A_1, A_2)$ is given completely when one of the two classes, e.~g., the first $A_1$ is known, because the second $A_2$ consists of all rational numbers not contained in $A_1$, and the characteristic property of such a first class lies in this % [File: 022.png] that if the number $a_1$ is contained in it, it also contains all numbers less than $a_1$. If now we compare two such first classes $A_1$, $B_1$ with each other, it may happen

1. That they are perfectly identical, i.~e., that every number contained in $A_1$ is also contained in $B_1$, and that every number contained in $B_1$ is also contained in $A_1$. In this case $A_2$ is necessarily identical with $B_2$, and the two cuts are perfectly identical, which we denote in symbols by $alpha=beta$ or $beta=alpha$.

But if the two classes $A_1$, $B_1$ are not identical, then there exists in the one, e.~g., in $A_1$, a number $a'_1=b'_2$ not contained in the other $B_1$ and consequently found in $B_2$; hence all numbers $b_1$ contained in $B_1$ are certainly less than this number $a'_1=b'_2$ and therefore all numbers $b_1$ are contained in $A_1$.

2. If now this number $a'_1$ is the only one in $A_1$ that is not contained in $B_1$, then is every other number $a_1$ contained in $A_1$ also contained in $B_1$ and is consequently $<a'_1$, i.~e., $a'_1$ is the greatest among all the numbers $a_1$, hence the cut $(A_1, A_2)$ is produced by the rational number $a=a'_1=b'_2$. Concerning the other cut $(B_1, B_2)$ we know already that all numbers $b_1$ in $B_1$ are also contained in $A_1$ and are less than the number $a'_1=b'_2$ which is contained in $B_2$; every other number $b_2$ contained in $B_2$ must, however, be greater than $b'_2$, for otherwise it would be less than $a'_1$, therefore contained in $A_1$ and hence in $B_1$; hence $b'_2$ is the least among all numbers contained in $B_2$, % [File: 023.png] and consequently the cut $(B_1, B_2)$ is produced by the same rational number $beta = b'_2 = a'_1 = alpha$. The two cuts are then only unessentially different.

3. If, however, there exist in $A_1$ at least two different numbers $a'_1 = b'_2$ and $a''_1 = b''_2$, which are not contained in $B_1$, then there exist infinitely many of them, because all the infinitely many numbers lying between $a'_1$ and $a''_1$ are obviously contained in $A_1$ (Section I, textsc{ii}) but not in $B_1$. In this case we say that the numbers $alpha$ and $beta$ corresponding to these two essentially different cuts $(A_1, A_2)$ and $(B_1, B_2)$ are textit{different}, and further that $alpha$ is textit{greater} than $beta$, that $beta$ is textit{less} than $alpha$, which we express in symbols by $alpha > beta$ as well as $beta < alpha$. It is to be noticed that this definition coincides completely with the one given earlier, when $alpha$, $beta$ are rational.

The remaining possible cases are these:

4. If there exists in $B_1$ one and only one number $b'_1 = a'_2$, that is not contained in $A_1$ then the two cuts $(A_1, A_2)$ and $(B_1, B_2)$ are only unessentially different and they are produced by one and the same rational number $alpha = a'_2 = b'_1 = beta$.

5. But if there are in $B_1$ at least two numbers which are not contained in $A_1$, then $beta > alpha$, $alpha < beta$.

As this exhausts the possible cases, it follows that of two different numbers one is necessarily the greater, the other the less, which gives two possibilities. A third case is impossible. This was indeed involved in the use of the textit{comparative} (greater, less) to designate % [File: 024.png] the relation between $alpha$, $beta$; but this use has only now been justified. In just such investigations one needs to exercise the greatest care so that even with the best intention to be honest he shall not, through a hasty choice of expressions borrowed from other notions already developed, allow himself to be led into the use of inadmissible transfers from one domain to the other.

If now we consider again somewhat carefully the case $alpha>beta$ it is obvious that the less number $beta$, if rational, certainly belongs to the class $A_1$; for since there is in $A_1$ a number $a'_1=b'_2$ which belongs to the class $B_2$, it follows that the number $beta$, whether the greatest number in $B_1$ or the least in $B_2$ is certainly $leqq a'_1$ and hence contained in $A_1$. Likewise it is obvious from $alpha>beta$ that the greater number $alpha$, if rational, certainly belongs to the class $B_2$, because $alpha geqq a'_1$. Combining these two considerations we get the following result: If a cut is produced by the number $alpha$ then any rational number belongs to the class $A_1$ or to the class $A_2$ according as it is less or greater than $alpha$; if the number $alpha$ is itself rational it may belong to either class.

From this we obtain finally the following: If $alpha>beta$, i.~e., if there are infinitely many numbers in $A_1$ not contained in $B_1$ then there are infinitely many such numbers that at the same time are different from $alpha$ and from $beta$; every such rational number $c$ is $<alpha$, because % [File: 025.png] it is contained in $A_1$ and at the same time it is $>beta$ because contained in $B_2$.

%V.

mysect{XIII}{V}{Continuity of the Domain of Real Numbers}

In consequence of the distinctions just established the system $mathfrak{R}$ of all real numbers forms a well-arranged domain of one dimension; this is to mean merely that the following laws prevail:

textsc{i.} If $alpha>beta$, and $beta>gamma$, then is also $alpha>gamma$. We shall say that the number $beta$ lies between $alpha$ and $gamma$.

textsc{ii.} If $alpha$, $gamma$ are any two different numbers, then there exist infinitely many different numbers $beta$ lying between $alpha$, $gamma$.

textsc{iii.} If $alpha$ is any definite number then all numbers of the system $mathfrak{R}$ fall into two classes $mathfrak{A}_1$ and $mathfrak{A}_2$ each of which contains infinitely many individuals; the first class $mathfrak{A}_1$ comprises all the numbers $alpha_1$ that are less than $alpha$, the second $mathfrak{A}_2$ comprises all the numbers $alpha_2$ that are greater than $alpha$; the number $alpha$ itself may be assigned at pleasure to the first class or to the second, and it is respectively the greatest of the first or the least of the second class. In each case the separation of the system $mathfrak{R}$ into the two classes $mathfrak{A}_1$, $mathfrak{A}_2$ is such that every number of first class $mathfrak{A}_1$ is smaller than every number of the second class $mathfrak{A}_2$ and we say that this separation is produced by the number $alpha$.

For brevity and in order not to weary the reader I suppress the proofs of these theorems which follow % [File: 026.png] immediately from the definitions of the previous section.

Beside these properties, however, the domain $mathfrak{R}$ possesses also textit{continuity}; i.~e., the following theorem is true:

textsc{iv.} label{EISViv}If the system $mathfrak{R}$ of all real numbers breaks up into two classes $mathfrak{A}_1$, $mathfrak{A}_2$ such that every number $alpha_1$ of the class $mathfrak{A}_1$ is less than every number $alpha_2$ of the class $mathfrak{A}_2$ then there exists one and only one number $alpha$ by which this separation is produced.

textit{Proof.} By the separation or the cut of $mathfrak{R}$ into $mathfrak{A}_1$ and $mathfrak{A}_2$ we obtain at the same time a cut $(A_1, A_2)$ of the system $R$ of all rational numbers which is defined by this that $A_1$ contains all rational numbers of the class $mathfrak{A}_1$ and $A_2$ all other rational numbers, i.~e., all rational numbers of the class $mathfrak{A}_2$. Let $alpha$ be the perfectly definite number which produces this cut $(A_1, A_2)$. If $beta$ is any number different from $alpha$, there are always infinitely many rational numbers $c$ lying between $alpha$ and $beta$. If $beta<alpha$, then $c<alpha$; hence $c$ belongs to the class $A_1$ and consequently also to the class $mathfrak{A}_1$, and since at the same time $beta<c$ then $beta$ also belongs to the same class $mathfrak{A}_1$, because every number in $mathfrak{A}_2$ is greater than every number $c$ in $mathfrak{A}_1$. But if $beta>alpha$, then is $c>alpha$; hence $c$ belongs to the class $A_2$ and consequently also to the class $mathfrak{A}_2$, and since at the same time $beta>c$, then $beta$ also belongs to the same class $mathfrak{A}_2$, because every number in $mathfrak{A}_1$ is less than every number $c$ in $mathfrak{A}_2$. Hence every number $beta$ different % [File: 027.png] from $alpha$ belongs to the class $mathfrak{A}_1$ or to the class $mathfrak{A}_2$ according as $beta<alpha$ or $beta>alpha$; consequently $alpha$ itself is either the greatest number in $mathfrak{A}_1$ or the least number in $mathfrak{A}_2$, i.~e., $alpha$ is one and obviously the only number by which the separation of $R$ into the classes $mathfrak{A}_1$, $mathfrak{A}_2$ is produced. Which was to be proved.

%VI.

mysect{XII}{VI}{Operations with Real Numbers} label{EISVI}

To reduce any operation with two real numbers $alpha$, $beta$ to operations with rational numbers, it is only necessary from the cuts $(A_1, A_2)$, $(B_1, B_2)$ produced by the numbers $alpha$ and $beta$ in the system $R$ to define the cut $(C_1, C_2)$ which is to correspond to the result of the operation, $gamma$. I confine myself here to the discussion of the simplest case, that of addition.

If $c$ is any rational number, we put it into the class $C_1$, provided there are two numbers one $a_1$ in $A_1$ and one $b_1$ in $B_1$ such that their sum $a_1+b_1 geqq c$; all other rational numbers shall be put into the class $C_2$. This separation of all rational numbers into the two classes $C_1$, $C_2$ evidently forms a cut, since every number $c_1$ in $C_1$ is less than every number $c_2$ in $C_2$. If both $alpha$ and $beta$ are rational, then every number $c_1$ contained in $C_1$ is $leqqalpha+beta$, because $a_1leqqalpha$, $b_1leqqbeta$, and therefore $a_1+b_1 leqqalpha+beta$; further, if there were contained in $C_2$ a number $c_2<alpha+beta$, hence $alpha+beta=c_2+p$, where $p$ is a positive rational number, then we should have [ c_2=(alpha- frac{1}{2} p)+(beta- frac{1}{2} p), ] % [File: 028.png] which contradicts the definition of the number $c_2$, because $alpha- frac{1}{2} p$ is a number in $A_1$, and $beta- frac{1}{2} p$ a number in $B_1$; consequently every number $c_2$ contained in $C_2$ is $geqqalpha+beta$. Therefore in this case the cut $(C_1, C_2)$ is produced by the sum $alpha+beta$. Thus we shall not violate the definition which holds in the arithmetic of rational numbers if in all cases we understand by the sum $alpha+beta$ of any two real numbers $alpha$, $beta$ that number $gamma$ by which the cut $(C_1, C_2)$ is produced. Further, if only one of the two numbers $alpha$, $beta$ is rational, e.~g., $alpha$, it is easy to see that it makes no difference with the sum $gamma=alpha+beta$ whether the number $alpha$ is put into the class $A_1$ or into the class $A_2$.

Just as addition is defined, so can the other operations of the so-called elementary arithmetic be defined, viz., the formation of differences, products, quotients, powers, roots, logarithms, and in this way we arrive at real proofs of theorems (as, e.~g., $sqrt{2}cdotsqrt{3} =sqrt{6}$), which to the best of my knowledge have never been established before. The excessive length that is to be feared in the definitions of the more complicated operations is partly inherent in the nature of the subject but can for the most part be avoided. Very useful in this connection is the notion of an textit{interval}, i.~e., a system $A$ of rational numbers possessing the following characteristic property: if $a$ and $a'$ are numbers of the system $A$, then are all rational numbers lying between $a$ and $a'$ contained in $A$. The system $R$ of all rational numbers, and also the two classes of any % [File: 029.png] cut are intervals. If there exist a rational number $a_1$ which is less and a rational number $a_2$ which is greater than every number of the interval $A$, then $A$ is called a finite interval; there then exist infinitely many numbers in the same condition as $a_1$ and infinitely many in the same condition as $a_2$; the whole domain $R$ breaks up into three parts $A_1$, $A$, $A_2$ and there enter two perfectly definite rational or irrational numbers $alpha_1$, $alpha_2$ which may be called respectively the lower and upper (or the less and greater) textit{limits} of the interval; the lower limit $alpha_1$ is determined by the cut for which the system $A_1$ forms the first class and the upper $alpha_2$ by the cut for which the system $A_2$ forms the second class. Of every rational or irrational number $alpha$ lying between $alpha_1$ and $alpha_2$ it may be said that it lies textit{within} the interval $A$. If all numbers of an interval $A$ are also numbers of an interval $B$, then $A$ is called a portion of $B$.

Still lengthier considerations seem to loom up when we attempt to adapt the numerous theorems of the arithmetic of rational numbers (as, e.~g., the theorem $(a+b)c=ac+bc$) to any real numbers. This, however, is not the case. It is easy to see that it all reduces to showing that the arithmetic operations possess a certain continuity. What I mean by this statement may be expressed in the form of a general theorem:

``If the number $lambda$ is the result of an operation performed on the numbers $alpha$, $beta$, $gamma$, $ldots$ and $lambda$ lies within the interval $L$, then intervals $A$, $B$, $C$, $ldots$ can be % [File: 030.png] taken within which lie the numbers $alpha$, $beta$, $gamma$, $ldots$ such that the result of the same operation in which the numbers $alpha$, $beta$, $gamma$, $ldots$ are replaced by arbitrary numbers of the intervals $A$, $B$, $C$, $ldots$ is always a number lying within the interval $L$.'' The forbidding clumsiness, however, which marks the statement of such a theorem convinces us that something must be brought in as an aid to expression; this is, in fact, attained in the most satisfactory way by introducing the ideas of textit{variable magnitudes}, textit{functions}, textit{limiting values}, and it would be best to base the definitions of even the simplest arithmetic operations upon these ideas, a matter which, however, cannot be carried further here.

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

mysect{XI}{VII}{Infinitesimal Analysis}

Here at the close we ought to explain the connection between the preceding investigations and certain fundamental theorems of infinitesimal analysis.

We say that a variable magnitude $x$ which passes through successive definite numerical values approaches a fixed limiting value $alpha$ when in the course of the process $x$ lies finally between two numbers between which $alpha$ itself lies, or, what amounts to the same, when the difference $x-alpha$ taken absolutely becomes finally less than any given value different from zero.

One of the most important theorems may be stated in the following manner: ``If a magnitude $x$ grows % [File: 031.png] continually but not beyond all limits it approaches a limiting value.''

I prove it in the following way. By hypothesis there exists one and hence there exist infinitely many numbers $alpha_2$ such that $x$ remains continually $<alpha_2$; I designate by $mathfrak{A}_2$ the system of all these numbers $alpha_2$, by $mathfrak{A}_1$ the system of all other numbers $alpha_1$; each of the latter possesses the property that in the course of the process $x$ becomes finally $geqq alpha_1$, hence every number $alpha_1$ is less than every number $alpha_2$ and consequently there exists a number $alpha$ which is either the greatest in $mathfrak{A}_1$ or the least in $mathfrak{A}_2$ (V, textsc{iv}). The former cannot be the case since $x$ never ceases to grow, hence $alpha$ is the least number in $mathfrak{A}_2$. Whatever number $alpha_1$ be taken we shall have finally $alpha_1<x<alpha$, i.~e., $x$ approaches the limiting value $alpha$.

This theorem is equivalent to the principle of continuity, i.~e., it loses its validity as soon as we assume a single real number not to be contained in the domain $mathfrak{R}$; or otherwise expressed: if this theorem is correct, then is also theorem textsc{iv}. in V. correct.

Another theorem of infinitesimal analysis, likewise equivalent to this, which is still oftener employed, may be stated as follows: ``If in the variation of a magnitude $x$ we can for every given positive magnitude $delta$ assign a corresponding position from and after which $x$ changes by less than $delta$ then $x$ approaches a limiting value.''

This converse of the easily demonstrated theorem % [File: 032.png] that every variable magnitude which approaches a limiting value finally changes by less than any given positive magnitude can be derived as well from the preceding theorem as directly from the principle of continuity. I take the latter course. Let $delta$ be any positive magnitude (i.~e., $delta>0$), then by hypothesis a time will come after which $x$ will change by less than $delta$, i.~e., if at this time $x$ has the value $a$, then afterwards we shall continually have $x>a-delta$ and $x<a+delta$. I now for a moment lay aside the original hypothesis and make use only of the theorem just demonstrated that all later values of the variable $x$ lie between two assignable finite values. Upon this I base a double separation of all real numbers. To the system $mathfrak{A}_2$ I assign a number $alpha_2$ (e.g., $a+delta$) when in the course of the process $x$ becomes finally $leqq alpha_2$; to the system $mathfrak{A}_1$ I assign every number not contained in $mathfrak{A}_2$; if $alpha_1$ is such a number, then, however far the process may have advanced, it will still happen infinitely many times that $x>alpha_2$. Since every number $alpha_1$ is less than every number $alpha_2$ there exists a perfectly definite number $alpha$ which produces this cut $(mathfrak{A}_1, mathfrak{A}_2)$ of the system $mathfrak{R}$ and which I will call the upper limit of the variable $x$ which always remains finite. Likewise as a result of the behavior of the variable $x$ a second cut $(mathfrak{B}_1, mathfrak{B}_2)$ of the system $mathfrak{R}$ is produced; a number $beta_2$ (e.~g., $a-delta$) is assigned to $mathfrak{B}_2$ when in the course of the process $x$ becomes finally $geqqbeta$; every other number $beta_2$, to be assigned to $mathfrak{B}_2$, has the property that $x$ is never % [File: 033.png] finally $geqq beta_2$; therefore infinitely many times $x$ becomes $<beta_2$; the number $beta$ by which this cut is produced I call the lower limiting value of the variable $x$. The two numbers $alpha$, $beta$ are obviously characterised by the following property: if $epsilon$ is an arbitrarily small positive magnitude then we have always finally $x < alpha + epsilon$ and $x > beta - epsilon$, but never finally $x < alpha - epsilon$ and never finally $x > beta + epsilon$. Now two cases are possible. If $alpha$ and $beta$ are different from each other, then necessarily $alpha > beta$, since continually $alpha_2 geqq beta_2$; the variable $x$ oscillates, and, however far the process advances, always undergoes changes whose amount surpasses the value $(alpha - beta) -2epsilon$ where $epsilon$ is an arbitrarily small positive magnitude. The original hypothesis to which I now return contradicts this consequence; there remains only the second case $alpha = beta$ since it has already been shown that, however small be the positive magnitude $epsilon$, we always have finally $x < alpha + epsilon$ and $x > beta - epsilon$, $x$ approaches the limiting value $alpha$, which was to be proved.

These examples may suffice to bring out the connection between the principle of continuity and infinitesimal analysis.

% [File: 034.png] %part*{THE NATURE AND MEANING OF NUMBERS}

% [File: 035.png] %[Blank Page] % [File: 036.png] newpage mychap{II}{THE NATURE AND MEANING OF NUMBERS} addcontentsline{toc}{section}{Prefaces} begin{center} {large PREFACE TO THE FIRST EDITION.} end{center}

In science nothing capable of proof ought to be accepted without proof. Though this demand seems so reasonable yet I cannot regard it as having been met even in the most recent methods of laying the foundations of the simplest science; viz., that part of logic which deals with the theory of numbers.footnote{Of the works which have come under my observation I mention the valuable textit{Lehrbuch der Arithmetik und Algebra} of E.~Schröder (Leipzig, 1873), which contains a bibliography of the subject, and in addition the memoirs of Kronecker and von Helmholtz upon the Number-Concept and upon Counting and Measuring (in the collection of philosophical essays published in honor of E.~Zeller, Leipzig, 1887). The appearance of these memoirs has induced me to publish my own views, in many respects similar but in foundation essentially different, which I formulated many years ago in absolute independence of the works of others.} In speaking of arithmetic (algebra, analysis) as a part of logic I mean to imply that I consider the number-concept entirely independent of the notions or intuitions of space and time, that I consider it an immediate result from the laws of thought. My answer to the problems propounded in the title of this paper is, then, briefly this: numbers are free creations of the human mind; they serve as a means of apprehending more easily and more sharply the difference of things. It is only through the purely logical process of building up the science of numbers and by thus acquiring % [File: 037.png] the continuous number-domain that we are prepared accurately to investigate our notions of space and time by bringing them into relation with this number-domain created in our mind.footnote{See Section III. of my memoir, textit{Continuity and Irrational Numbers} (Braunschweig, 1872), translated at pages pageref{EISIII} et seq.of the present volume.} If we scrutinise closely what is done in counting an aggregate or number of things, we are led to consider the ability of the mind to relate things to things, to let a thing correspond to a thing, or to represent a thing by a thing, an ability without which no thinking is possible. Upon this unique and therefore absolutely indispensable foundation, as I have already affirmed in an announcement of this paper,footnote{Dirichlet's textit{Vorlesungen über Zahlentheorie}, third edition, 1879, § 163, note on page 470.} must, in my judgment, the whole science of numbers be established. The design of such a presentation I had formed before the publication of my paper on textit{Continuity}, but only after its appearance and with many interruptions occasioned by increased official duties and other necessary labors, was I able in the years 1872 to 1878 to commit to paper a first rough draft which several mathematicians examined and partially discussed with me. It bears the same title and contains, though not arranged in the best order, all the essential fundamental ideas of my present paper, in which they are more carefully elaborated. As such main points I mention here the sharp distinction between finite and infinite (64), the notion of the number [textit{Anzahl}] of things (161), the % [File: 038.png] proof that the form of argument known as complete induction (or the inference from $n$ to $n + 1$) is really conclusive (59), (60), (80), and that therefore the definition by induction (or recursion) is determinate and consistent (126).

This memoir can be understood by any one possessing what is usually called good common sense; no technical philosophic, or mathematical, knowledge is in the least degree required. But I feel conscious that many a reader will scarcely recognise in the shadowy forms which I bring before him his numbers which all his life long have accompanied him as faithful and familiar friends; he will be frightened by the long series of simple inferences corresponding to our step-by-step understanding, by the matter-of-fact dissection of the chains of reasoning on which the laws of numbers depend, and will become impatient at being compelled to follow out proofs for truths which to his supposed inner consciousness seem at once evident and certain. On the contrary in just this possibility of reducing such truths to others more simple, no matter how long and apparently artificial the series of inferences, I recognise a convincing proof that their possession or belief in them is never given by inner consciousness but is always gained only by a more or less complete repetition of the individual inferences. I like to compare this action of thought, so difficult to trace on account of the rapidity of its performance, with the action which an accomplished reader performs % [File: 039.png] in reading; this reading always remains a more or less complete repetition of the individual steps which the beginner has to take in his wearisome spelling-out; a very small part of the same, and therefore a very small effort or exertion of the mind, is sufficient for the practised reader to recognise the correct, true word, only with very great probability, to be sure; for, as is well known, it occasionally happens that even the most practised proof-reader allows a typographical error to escape him, i.~e., reads falsely, a thing which would be impossible if the chain of thoughts associated with spelling were fully repeated. So from the time of birth, continually and in increasing measure we are led to relate things to things and thus to use that faculty of the mind on which the creation of numbers depends; by this practice continually occurring, though without definite purpose, in our earliest years and by the attending formation of judgments and chains of reasoning we acquire a store of real arithmetic truths to which our first teachers later refer as to something simple, self-evident, given in the inner consciousness; and so it happens that many very complicated notions (as for example that of the number [textit{Anzahl}] of things) are erroneously regarded as simple. In this sense which I wish to express by the word formed after a well-known saying textit{textgreek{as eig{} oR{} asa njrwpos as rijmhtia zai}}, %$alphaepsiloniota;omicron;alphanuthetarhoomegapiomicronvarsigma;alpharhoiotathetamuetatauiotazetaepsiloniota$, I hope that the following pages, as an attempt to establish the science of numbers upon a uniform foundation will find a generous % [File: 040.png] welcome and that other mathematicians will be led to reduce the long series of inferences to more moderate and attractive proportions.

In accordance with the purpose of this memoir I restrict myself to the consideration of the series of so-called natural numbers. In what way the gradual extension of the number-concept, the creation of zero, negative, fractional, irrational and complex numbers are to be accomplished by reduction to the earlier notions and that without any introduction of foreign conceptions (such as that of measurable magnitudes, which according to my view can attain perfect clearness only through the science of numbers), this I have shown at least for irrational numbers in my former memoir on textit{Continuity} (1872); in a way wholly similar, as I have already shown in Section~III. of that memoir,footnote{Pages pageref{EISIII} et seq.of the present volume.} may the other extensions be treated, and I propose sometime to present this whole subject in systematic form. From just this point of view it appears as something self-evident and not new that every theorem of algebra and higher analysis, no matter how remote, can be expressed as a theorem about natural numbers,---a declaration I have heard repeatedly from the lips of Dirichlet. But I see nothing meritorious--and this was just as far from Dirichlet's thought---in actually performing this wearisome circumlocution and insisting on the use and recognition of no other than rational numbers. On the contrary, % [File: 041.png] the greatest and most fruitful advances in mathematics and other sciences have invariably been made by the creation and introduction of new concepts, rendered necessary by the frequent recurrence of complex phenomena which could be controlled by the old notions only with difficulty. On this subject I gave a lecture before the philosophic faculty in the summer of 1854 on the occasion of my admission as privat-docent in Göttingen. The scope of this lecture met with the approval of Gauss; but this is not the place to go into further detail.

Instead of this I will use the opportunity to make some remarks relating to my earlier work, mentioned above, on textit{Continuity and Irrational Numbers}. The theory of irrational numbers there presented, wrought out in the fall of 1853, is based on the phenomenon (Section IV.)footnote{Pages pageref{EISIV} et seq.of the present volume.} occurring in the domain of rational numbers which I designate by the term cut [textit{Schnitt}] and which I was the first to investigate carefully; it culminates in the proof of the continuity of the new domain of real numbers (Section V., textsc{iv}.).footnote{Page pageref{EISViv} of the present volume.} It appears to me to be somewhat simpler, I might say easier, than the two theories, different from it and from each other, which have been proposed by Weierstrass and G.~Cantor, and which likewise are perfectly rigorous. It has since been adopted without essential modification by U.~Dini in his textit{Fondamenti per la teorica delle % [File: 042.png] funzioni di variabili reali} (Pisa, 1878); but the fact that in the course of this exposition my name happens to be mentioned, not in the description of the purely arithmetic phenomenon of the cut but when the author discusses the existence of a measurable quantity corresponding to the cut, might easily lead to the supposition that my theory rests upon the consideration of such quantities. Nothing could be further from the truth; rather have I in Section III.footnote{Pages pageref{EISIII} et seq.of the present volume.} of my paper advanced several reasons why I wholly reject the introduction of measurable quantities; indeed, at the end of the paper I have pointed out with respect to their existence that for a great part of the science of space the continuity of its configurations is not even a necessary condition, quite aside from the fact that in works on geometry arithmetic is only casually mentioned by name but is never clearly defined and therefore cannot be employed in demonstrations. To explain this matter more clearly I note the following example: If we select three non-collinear points $A$, $B$, $C$ at pleasure, with the single limitation that the ratios of the distances $AB$, $AC$, $BC$ are algebraic numbers,footnote{Dirichlet's textit{Vorlesungen über Zahlentheorie}, §~159 of the second edition, §~160 of the third.} and regard as existing in space only those points $M$, for which the ratios of $AM$, $BM$, $CM$ to $AB$ are likewise algebraic numbers, then is the space made up of the points $M$, as is easy to see, everywhere discontinuous; % [File: 043.png] but in spite of this discontinuity, and despite the existence of gaps in this space, all constructions that occur in Euclid's textit{Elements}, can, so far as I can see, be just as accurately effected as in perfectly continuous space; the discontinuity of this space would not be noticed in Euclid's science, would not be felt at all. If any one should say that we cannot conceive of space as anything else than continuous, I should venture to doubt it and to call attention to the fact that a far advanced, refined scientific training is demanded in order to perceive clearly the essence of continuity and to comprehend that besides rational quantitative relations, also irrational, and besides algebraic, also transcendental quantitative relations are conceivable. All the more beautiful it appears to me that without any notion of measurable quantities and simply by a finite system of simple thought-steps man can advance to the creation of the pure continuous number-domain; and only by this means in my view is it possible for him to render the notion of continuous space clear and definite.

The same theory of irrational numbers founded upon the phenomenon of the cut is set forth in the textit{Introduction à la théorie des fonctions d'une variable} by J.~Tannery (Paris, 1886). If I rightly understand a passage in the preface to this work, the author has thought out his theory independently, that is, at a time when not only my paper, but Dini's textit{Fondamenti} mentioned in the same preface, was unknown to him. % [File: 044.png] This agreement seems to me a gratifying proof that my conception conforms to the nature of the case, a fact recognised by other mathematicians, e.~g., by Pasch in his textit{Einleitung in die Differential- und Integralrechnung} (Leipzig, 1883). But I cannot quite agree with Tannery when he calls this theory the development of an idea due to J.~Bertrand and contained in his textit{Traité d'arithmétique}, consisting in this that an irrational number is defined by the specification of all rational numbers that are less and all those that are greater than the number to be defined. As regards this statement which is repeated by Stolz---apparently without careful investigation---in the preface to the second part of his textit{Vorlesungen über allgemeine Arithmetik} (Leipzig, 1886), I venture to remark the following: That an irrational number is to be considered as fully defined by the specification just described, this conviction certainly long before the time of Bertrand was the common property of all mathematicians who concerned themselves with the notion of the irrational. Just this manner of determining it is in the mind of every computer who calculates the irrational root of an equation by approximation, and if, as Bertrand does exclusively in his book, (the eighth edition, of the year 1885, lies before me,) one regards the irrational number as the ratio of two measurable quantities, then is this manner of determining it already set forth in the clearest possible way in the celebrated definition which Euclid gives of the equality % [File: 045.png] of two ratios (textit{Elements}, V., 5). This same most ancient conviction has been the source of my theory as well as that of Bertrand and many other more or less complete attempts to lay the foundations for the introduction of irrational numbers into arithmetic. But though one is so far in perfect agreement with Tannery, yet in an actual examination he cannot fail to observe that Bertrand's presentation, in which the phenomenon of the cut in its logical purity is not even mentioned, has no similarity whatever to mine, inasmuch as it resorts at once to the existence of a measurable quantity, a notion which for reasons mentioned above I wholly reject. Aside from this fact this method of presentation seems also in the succeeding definitions and proofs, which are based on the postulate of this existence, to present gaps so essential that I still regard the statement made in my paper (Section VI.),footnote

{Pages pageref{EISVI} et seq.of this volume.}

that the theorem $sqrt{2}cdotsqrt{3}=sqrt{6}$ 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.