The Joy of Sets: Fundamentals of Contemporary Set Theory

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Springer Science & Business Media, Jun 24, 1994 - Mathematics - 194 pages
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This book provides an account of those parts of contemporary set theory of direct relevance to other areas of pure mathematics. The intended reader is either an advanced-level mathematics undergraduate, a beginning graduate student in mathematics, or an accomplished mathematician who desires or needs some familiarity with modern set theory. The book is written in a fairly easy-going style, with minimal formalism. In Chapter 1, the basic principles of set theory are developed in a 'naive' manner. Here the notions of 'set', 'union', 'intersection', 'power set', 'rela tion', 'function', etc., are defined and discussed. One assumption in writing Chapter 1 has been that, whereas the reader may have met all of these 1 concepts before and be familiar with their usage, she may not have con sidered the various notions as forming part of the continuous development of a pure subject (namely, set theory). Consequently, the presentation is at the same time rigorous and fast.
 

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Contents

Naive Set Theory
1
12 Operations on Sets
4
13 Notation for Sets
6
14 Sets of Sets
7
15 Relations
10
16 Functions
12
17 WellOrderings and Ordinals
16
18 Problems
25
42 Closed Unbounded Sets
103
43 Stationary Sets and Regressive Functions
106
44 Trees
109
45 Extensions of Lebesgue Measure
113
46 A Result About the GCH
116
The Axiom of Constructibility
120
52 The Constructible Hierarchy
123
53 The Axiom of Constructibility
124

The ZermeloPraenkel Axioms
29
21 The Language of Set Theory
30
22 The Cumulative Hierarchy of Sets
35
23 The ZermeloFraenkel Axioms
40
24 Classes
46
25 Set Theory as an Axiomatic Theory
50
26 The Recursion Principle
51
27 The Axiom of Choice
56
28 Problems
63
Ordinal and Cardinal Numbers
66
32 Addition of Ordinals
68
33 Multiplication of Ordinals
69
34 Sequences of Ordinals
71
35 Ordinal Exponentiation
74
36 Cardinality Cardinal Numbers
75
37 Arithmetic of Cardinal Numbers
82
38 Regular and Singular Cardinals
88
39 Cardinal Exponentiation
91
310 Inaccessible Cardinals
95
311 Problems
98
Topics in Pure Set Theory
101
54 The Consistency of V L
127
55 Use of the Axiom of Constructibility
128
Independence Proofs in Set Theory
130
63 The BooleanValued Universe
133
64 VB and V
136
65 BooleanValued Sets and Independence Proofs
137
66 The Nonprovability of the CH
139
NonWeilFounded Set Theory
143
71 SetMembership Diagrams
145
72 The AntiFoundation Axiom
151
73 The Solution Lemma
156
74 Inductive Definitions Under AFA
159
75 Graphs and Systems
163
76 Proof of the Solution Lemma
168
77 CoInductive Definitions
169
78 A Model of ZF +AFA
173
Bibliography
185
Glossary of Symbols
186
Index
189
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Page 185 - London, 1987. [3] JL Bell, Boolean- Valued Models and Independence Proofs in Set Theory, Oxford University Press, London, 1977.

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About the author (1994)

Born in England in 1947 and living in America since 1987, Keith Devlin has written more than 20 books and numerous research articles on various elements of mathematics. From 1983 to 1989, he wrote a column on for the Manchester (England) Guardian. The collected columns are published in All the Math That's Fit to Print (1994) and cover a wide range of topics from calculating travel expenses to calculating pi. His book Logic and Information (1991) is an introduction to situation theory and situation semantics for mathematicians. Co-author of the PBS Nova episode "A Mathematical Mystery Tour," he is also the author of Devlin's Angle, a column on the Mathematical Association of America's electronic journal. Devlin lives in California, where he is dean of the school of science at Saint Mary's College in Morgana. He is currently studying the use of mathematics to analyze communication and information flow in the workplace.

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