Dalen / Doets / de Swart Sets: Naïve, Axiomatic and Applied

A Basic Compendium with Exercises for Use in Set Theory for Non Logicians, Working and Teaching Mathematicians and Students
1. Auflage 2014
ISBN: 978-1-4831-5039-0
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 360 Seiten, Web PDF

Sets: Naïve, Axiomatic and Applied is a basic compendium on naïve, axiomatic, and applied set theory and covers topics ranging from Boolean operations to union, intersection, and relative complement as well as the reflection principle, measurable cardinals, and models of set theory. Applications of the axiom of choice are also discussed, along with infinite games and the axiom of determinateness. Comprised of three chapters, this volume begins with an overview of naïve set theory and some important sets and notations. The equality of sets, subsets, and ordered pairs are considered, together with equivalence relations and real numbers. The next chapter is devoted to axiomatic set theory and discusses the axiom of regularity, induction and recursion, and ordinal and cardinal numbers. In the final chapter, applications of set theory are reviewed, paying particular attention to filters, Boolean algebra, and inductive definitions together with trees and the Borel hierarchy. This book is intended for non-logicians, students, and working and teaching mathematicians.

Autoren/Hrsg.

Dalen, D. van

Doets, H. C.

de Swart, H.

Sneddon, I. N.

Inhaltsverzeichnis

1;Front Cover;1
2;Sets: Naïve, Axiomatic and Applied;4
3;Copyright Page;5
4;Table of Contents;6
5;Preface;10
6;Acknowledgements ;14
7;Introduction ;16
8;Chapter 1. Naive Set Theory;20
8.1;1. Some important sets and notations;20
8.2;2. Equality of sets ;22
8.3;3. Subsets ;23
8.4;4. The Naive Comprehension principle and the empty set ;25
8.5;5. Union, intersection and relative complement Complement, de Morgan's laws;28
8.6;6. Power set ;37
8.7;7. Unions and intersections of families;40
8.8;8. Ordered pairs ;48
8.9;9. Cartesian product ;51
8.10;10. Relations ;55
8.11;11. Equivalence relations ;60
8.12;12. Real numbers ;70
8.13;13. Functions (mappings) ;75
8.14;14. Orderings ;97
8.15;15. Equivalence (cardinality) ;111
8.16;16. Finite and infinite ;128
8.17;17. Denumerable sets ;135
8.18;18. Uncountable sets;146
8.19;19. The paradoxes ;151
8.20;20. The set theory of Zermelo-Fraenkel (ZF) ;155
8.21;21. Peano's arithmetic ;167
9;Chapter 2. Axiomatic Set Theory;171
9.1;1. The axiom of regularity;171
9.2;2. Induction and Recursion;175
9.3;3. Ordinal numbers;184
9.4;4. The cumulative hierarchy;187
9.5;5. Ordinal arithmetic;193
9.6;6. Normal operations ;199
9.7;7. The reflection principle ;206
9.8;8. Initial numbers ;209
9.9;9. The axiom of choice;213
9.10;10. Cardinal numbers ;223
9.11;11. Models ;235
9.12;12. Measurable cardinals ;248
10;Chapter 3. Applications ;258
10.1;1. Filters ;258
10.2;2. Boolean algebra ;261
10.3;3. Order types ;272
10.4;4. Inductive definitions ;281
10.5;5. Applications of the axiom of choice ;287
10.6;6. The Borel hierarchy ;294
10.7;7. Trees ;316
10.8;8. The axiom, of Determinateness (AD) ;332
11;Appendix ;342
12;Symbols ;347
13;Literature ;350
14;Index ;352
15;Other Titles in the Series ;362