Essays on the Theory of Numbers, Volume 1

Richard Dedekind

Courier Corporation, Jan 1, 1963 - Mathematics - 115 pages

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.

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
• 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
• 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
• 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

Bibliographic information

• Title: Essays on the Theory of Numbers, Volume 1; Dover Books on Mathematics Series; Volume 1010 of Dover Books on Science; Dover Books; Dover books on advanced mathematics; Dover books on intermediate and advanced mathematics; Dover edition; Essays on the Theory of Numbers
• Author: Richard Dedekind
• Edition: unabridged, reprint
• Publisher: Courier Corporation, 1963
• ISBN: 0486210103, 9780486210100
• Length: 115 pages
• Subjects: Mathematics / General; Mathematics / Number Theory