Theory of Sets

Contents......Page 4
Introduction......Page 6
1. A First Classification of Sets......Page 7
2. Three Remarkable Examples of Enumerable Sets......Page 8
3. Subset, Sum, and Intersection of Sets; in Particular, of Enumerable Sets......Page 11
4. An Example of a Nonenumerable Set......Page 15
1. Extensions of the Number Concept......Page 18
2. Equivalence of Sets......Page 19
3. Cardinal Numbers......Page 23
4. Introductory Remarks Concerning the Scale of Cardinal Numbers......Page 26
5. F. Bernstein\'s Equivalence-Theorem......Page 28
6. The Sum of Two Cardinal Numbers......Page 31
7. The Product of Two Cardinal Numbers......Page 34
8. The Sum of Arbitrarily Many Cardinal Numbers......Page 38
9. The Product of Arbitrarily Many Cardinal Numbers......Page 43
10. The Power......Page 47
11. Some Examples of the Evaluation of Powers......Page 53
1. Definition of Ordered Set......Page 58
2. Similarity and Order Type......Page 61
3. The Sum of Order Types......Page 64
4. The Product of Two Order Types......Page 67
5. Power of Type Classes......Page 72
6. Dense Sets......Page 75
7. Continuous Sets......Page 79
1. Definition of Well-ordering and of Ordinal Number......Page 85
2. Addition of Arbitrarily Many, and Multiplication of Two, Ordinal Numbers......Page 87
3. Subsets and Similarity Mappings of Well-ordered Sets......Page 88
4. The Comparison of Ordinal Numbers......Page 91
5. Sequences of Ordinal Numbers......Page 95
6. Operating with Ordinal Numbers......Page 99
7. The Sequence of Ordinal Numbers, and Transfinite Induction......Page 104
8. The Product of Arbitrarily Many Ordinal Numbers......Page 106
9. Powers of Ordinal Numbers......Page 110
10. Polynomials in Ordinal Numbers......Page 113
11. The Well-ordering Theorem......Page 116
12. An Application of the Well-ordering Theorem......Page 121
13. The Well-ordering of Cardinal Numbers......Page 125
14. Further Rules of Operation for Cardinal Numbers. Order Type of Number Classes......Page 127
15. Ordinal Numbers and Sets of Points......Page 132
Concluding Remarks......Page 141
Bibliography......Page 145
Key to Symbols......Page 146
Index......Page 147