This introduction to the theory of sets employs the discoveries of Cantor, Russell, Weierstrass, Zermelo, Bernstein, Dedekind, and others. It analyzes concepts and principles, offering numerous examples. Topics include the rudiments of set theory, arbitrary sets and their cardinal numbers, ordered sets and their order types, and well-ordered sets and their ordinal numbers. 1950 edition.

Unabridged republication of the 1950 Dover edition.

Table of Contents for Theory of Sets

INTRODUCTION
CHAPTER I. THE RUDIMENTS OF SET THEORY
1. A First Classification of Sets
2. Three Remarkable Examples of Enumerable Sets
3. "Subset, Sum, and Intersection of Sets; in Particular, of Enumerable Sets"
4. An Example of a Nonenumerable Set
CHAPTER II. ARBITRARY SETS AND THEIR CARDINAL NUMBERS
1. Extensions of the Number Concept
2. Equivalence of Sets
3. Cardinal Numbers
4. Introductory Remarks Concerning the Scale of Cardinal Numbers
5. F. Bernstein's Equivalence-Theorem
6. The Sum of Two Cardinal Numbers
7. The Product of Two Cardinal Numbers
8. The Sum of Arbitrarily Many Cardinal Numbers
9. The Product of Arbitrarily Many Cardinal Numbers
10. The Power
11. Some Examples of the Evaluation of Powers
CHAPTER III. ORDERED SETS AND THEIR ORDER TYPES
1. Definition of Ordered Set
2. Similarity and Order Type
3. The Sum of Order Types
4. The Product of Two Order Types
5. Power of Type Classes
6. Dense Sets
7. Continuous Sets
CHAPTER IV. WELL-ORDERED SETS AND THEIR ORDINAL NUMBERS
1. Definition of Well-ordering and of Ordinal Number
2. "Addition of Arbitrarily Many, and Multiplication of Two, Ordinal Numbers"
3. Subsets and Similarity Mappings of Well-ordered Sets
4. The Comparison of Ordinal Numbers
5. Sequences of Ordinal Numbers
6. Operating with Ordinal Numbers
7. "The Sequence of Ordinal Numbers, and Transfinite Induction"
8. The Product of Arbitrarily Many Ordinal Numbers
9. Powers of Ordinal Numbers
10. Polynomials in Ordinal Numbers
11. The Well-ordering Theorem
12. An Application of the Well-ordering Theorem
13. The Well-ordering of Cardinal Numbers
14. Further Rules of Operation for Cardinal Numbers. Order Type of Number Classes
15. Ordinal Numbers and Sets of Points
CONCLUDING REMARKS
BIBLIOGRAPHY
KEY TO SYMBOLS
INDEX

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