Exceptionally well written. - School Science and Mathematics A very fine book. - Mathematics Teacher Of real service to logicians and philosophers who have hitherto had no access to a concise and accurate introduction to the general theory of sets. - Philosophical ReviewThis is the clearest and simplest introduction yet written to the theory of sets. Making use of the discoveries of Cantor, Russell, Weierstrass, Zermelo, Bernstein, Dedekind, and other mathematicians, it analyses concepts and principles and offers innumerable examples. Its emphasis is on fundamentals and the presentation is easily comprehensible to readers with some college algebra. But special subdivisions, such as the theory of sets of points, are considered.
The contents include rudiments (first classifications, subsets, sums, intersection of sets, nonenumerable sets, etc.); arbitrary sets and their cardinal numbers (extensions of number concept, equivalence of sets, sums and products of two and many cardinal numbers, etc.); ordered sets and their order types; and well-ordered sets and their ordinal numbers (addition and multiplication of ordinal numbers, transfinite induction, products and powers of ordinal numbers, well-ordering theorem, well-ordering of cardinal and ordinal numbers, etc.).
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Series: Dover Books on Mathematics
28 March 2003
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