Essays on the Theory of Numbers

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 "e;Vooster "e; V. Beman.

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 "e;Vooster "e; V. Beman.

I. Continuity and Irrational Numbers
Preface
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
Prefaces
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