Essays on the Theory of Numbers, Volume 1This 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. |
Contents
Properties of Rational Numbers | 3 |
Comparison of the Rational Numbers with the Points of a Straight Line | 6 |
Continuity of the Straight Line | 8 |
Creation of Irrational Numbers V Continuity of the Domain of Real Numbers | 12 |
Operations with Real Numbers | 21 |
Infinitesimal Analysis | 24 |
THE NATURE AND MEANING OF NUMBERS | 29 |
Prefaces PAGE I | 32 |
The Finite and Infinite | 63 |
Simply Infinite Systems Series of Natural Numbers | 67 |
Greater and Less Numbers | 70 |
31 | 74 |
44 | 75 |
Finite and Infinite Parts of the NumberSeries | 81 |
Definition of a Transformation of the NumberSeries by Induction | 83 |
The Class of Simply Infinite Systems | 92 |
6 | 40 |
8 | 43 |
12 | 47 |
19 | 49 |
Systems of Elements II Transformation of a System | 50 |
Similarity of a Transformation Similar Systems IV Transformation of a System in Itself | 56 |
Addition of Numbers | 96 |
Multiplication of Numbers | 101 |
Involution of Numbers 83 92 96 ΙΟΙ | 104 |
Number of the Elements of a Finite System | 105 |