Set theory is a funny discipline. For ages and ages mathematics has managed without set theory, but nowadays one gets from the average textbook the impression that set theory is absolutely indispensable. Even texts for high schools (not to mention nursery schools) start with sets, unions, intersections, etc.
Among the professional mathematicians there are some who claim that "there are no things but sets" How do these extremists justify their opinion? We will try to unearth some of the motivation for a belief in the supremacy of set theory.
Author(s): D. van Dalen, H.C.Doets, H. de Swart
Series: International Series in Pure and Applied Mathematics 106
Publisher: Pergamon Press
Year: 1978
Language: English
Pages: 354
Sets: Naive, Axiomatic and Applied (1978) ......Page 1
Preface ......Page 4
Acknowledgements ......Page 8
Introduction ......Page 9
1. Some important sets and notations ......Page 12
2. Equality of sets ......Page 14
3. Subsets ......Page 15
4. The naive comprehension principle and the empty set ......Page 17
5. Union, intersection and relative complement ......Page 20
6. Power set ......Page 29
7. Unions and intersections of families ......Page 32
8. Ordered pairs ......Page 40
9. Cartesian product ......Page 43
10. Relations ......Page 47
11. Equivalence relations ......Page 52
12. Real numbers ......Page 62
13. Functions (mappings) ......Page 67
14. Orderings ......Page 89
15. Equivalence (cardinality) ......Page 103
16. Finite and infinite ......Page 120
17. Denumerable sets ......Page 127
18. Uncountable sets ......Page 138
19. The paradoxes ......Page 143
20. The set theory of Zermelo-Fraenkel (ZF) ......Page 147
21. Peano's Arithmetic ......Page 159
1. The axiom of regularity ......Page 163
2. Induction and recursion ......Page 167
3. Ordinal numbers ......Page 176
4. The cumulative hierarchy ......Page 179
5. Ordinal arithmetic ......Page 185
6. Normal operations ......Page 191
7. The reflection-principle ......Page 198
8. Initial numbers ......Page 201
9. The axiom of choice ......Page 205
10. Cardinal numbers ......Page 215
11. Models ......Page 227
12. Measurable cardinals ......Page 240
1. Filters ......Page 250
2. Boolean algebras ......Page 253
3. Order types ......Page 264
4. Inductive definitions ......Page 273
5. Applications of the axiom of choice ......Page 279
6. The Borel hierarchy ......Page 286
7. Trees ......Page 308
8. The axiom of determinateness ......Page 324
Appendix ......Page 334
Symbols ......Page 339
Literature ......Page 342
Index ......Page 345
Other titles in the "International series in pure and applied mathematics" ......Page 353
