Essays on the Theory of Numbers

Essays on the Theory of Numbers

by Richard Dedekind


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This volume contains the two most important essays on the logical foundations of the number system by the famous German mathematician J. W. R. Dedekind. The first presents Dedekind's theory of the irrational number-the Dedekind cut idea-perhaps the most famous of several such theories created in the 19th century to give a precise meaning to irrational numbers, which had been used on an intuitive basis since Greek times. This paper provided a purely arithmetic and perfectly rigorous foundation for the irrational numbers and thereby a rigorous meaning of continuity in analysis.
The second essay is an attempt to give a logical basis for transfinite numbers and properties of the natural numbers. It examines the notion of natural numbers, the distinction between finite and transfinite (infinite) whole numbers, and the logical validity of the type of proof called mathematical or complete induction.
The contents of these essays belong to the foundations of mathematics and will be welcomed by those who are prepared to look into the somewhat subtle meanings of the elements of our number system. As a major work of an important mathematician, the book deserves a place in the personal library of every practicing mathematician and every teacher and historian of mathematics. Authorized translations by "Vooster " V. Beman.

Product Details

ISBN-13: 9780486210100
Publisher: Dover Publications
Publication date: 06/01/1963
Series: Dover Books on Mathematics
Pages: 128
Product dimensions: 5.00(w) x 8.00(h) x 0.28(d)

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Essays on the Theory of Numbers

By Richard Dedekind, Wooster Woodruff Beman

Dover Publications, Inc.

Copyright © 1963 Dover Publications, Inc.
All rights reserved.
ISBN: 978-0-486-21010-0



The development of the arithmetic of rational numbers is here presupposed, but still I think it worth while to call attention to certain important 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, which I shall denote by R, possesses first of all a completeness and self-containedness which I have designated in another place as characteristic of a 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 greater than b, b less than a; this is also indicated by the symbols a > b, b<a. As in the latter case ba has a positive value it follows that b>a, a<b. In regard to these two ways in which two numbers may differ the following laws will hold:

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.

II. If a, c are two different numbers, there are infinitely many different numbers lying between a, c.

II. If a is any definite number, then all numbers of the system R fall into two classes, A1 and A2 each of which contains infinitely many individuals; the first class A1 comprises all numbers a1 that are < a, the second class A2 comprises all numbers a2 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 A1, A2 is such that every number of the first class A1 is less than every number of the second class A2.



The above-mentioned properties of rational numbers recall the corresponding relations of position of 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 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:

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.

II. If p, r are two different points, then there always exist infinitely many points that lie between p and r.

III. If p is a definite point in L, then all points in L fall into two classes, P1, P2, each of which contains infinitely many individuals; the first class P1 contains all the points p1, that lie to the left of p, and the second class P2 contains all the points p2 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 P1, P2, is of such a character that every point of the first class P1 lies to the left of every point of the second class P2.

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 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 I, II, III of the previous Section correspond completely the laws I, II, III of the present.



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 op 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 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 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 of the same kind. 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 all continuous domains. By vague remarks upon the unbroken connection in 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 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 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.



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 a1 of the first class A1 is less than every number a2 of the second class A2; the number a is either the greatest number of the class A1 or the least number of the class A2. If now any separation of the system R into two classes A1, A2, is given which possesses only this characteristic property that every number a1 in A1 is less than every number a2 in A2, then for brevity we shall call such a separation a cut [Schnitt] and designate it by (A1, A2). 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, 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 λ such that

λ2< D < (λ+1)2.

If we assign to the second class A2, every positive rational number a2 whose square is >D, to the first class A1 all other rational numbers a1, this separation forms a cut (A1, A2), i. e., every number a1 is less than every number a2. For if a1 = 0, or is negative, then on that ground a1 is less than any number a2, because, by definition, this last is positive; if a1 is positive, then is its square D, and hence a1 is less than any positive number a2 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. 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

t2 - Du2 = 0,

and we may assume that u is the 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

λu < t < (λ + 1)u,

the number u' = t — λu is a positive integer certainly less than u. If further we put

t' = Du - λt,

t' is likewise a positive integer, and we have

t'2 - Du'2 = (λ2 - D) (t2 - Du2) = 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 A1 a greatest, nor in the class A2 a least number. For if we put



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Table of Contents

I. Continuity and Irrational Numbers
1. Properties of Rational Numbers
2. Comparison of the Rational Numbers with the Points of a Straight Line
3. Continuity of the Straight Line
4. Creation of Irrational Numbers
5. Continuity of the Domain of Real Numbers
6. Operations with Real Numbers
7. Infinitesimal Analysis
II. The Nature and Meaning of Numbers
1. Systems of Elements
2. Transformation of a System
3. Similarity of a Transformation. Similar Systems
4. Transformation of a System in Itself
5. The Finite and Infinite
6. Simply Infinite Systems. Series of Natural Numbers
7. Greater and Less Numbers
8. Finite and Infinite Parts of the Number-Series
9. Definition of a Transformation of the Number-Series by Induction
10. The Class of Simply Infinite Systems
11. Addition of Numbers
12. Multiplication of Numbers
13. Involution of Numbers
14. Number of the Elements of a Finite System

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