The Joy of Sets: Fundamentals of Contemporary Set TheoryThis 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
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 |
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| 71 | |
| 74 | |
| 75 | |
| 82 | |
| 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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Common terms and phrases
abbreviates assume Axiom of Choice Axiom of Constructibility Axiom of Foundation Axiom of Infinity Axiom of Replacement Axiom of Subset axioms of set basic bijection binary relation bisimulation boolean algebra boolean-valued chM(a Clearly cofinal concept consistent constructible set theory Corollary countable define denote easily seen elements equivalence existence fact finite sets fixed-point formula of LAST Fraenkel function f graph Hence inaccessible cardinal induction infinite cardinal isomorphic Let f limit ordinal logic mathematical maximal means non-well-founded sets nonempty sets notation notion operation ordered pair ordinal number system poset power set proper class prove recursion principle result Second edition Set Axiom set-theoretic Solution Lemma Subset Selection successor ordinal Suppose system map system of equations tagged Theorem theory of sets top node uncountable unique decoration well-founded well-ordering woset Zermelo hierarchy Zermelo-Fraenkel set theory ZFC axioms ZFCA
Popular passages
Page 61 - Lemma 3.3.1 (Zorn's lemma) Let (P, <) be a nonempty partially ordered set such that every chain in P has an upper bound in P. Then P has a maximal element.
Page 6 - This is because some of the dependencies (dependencies having the same target role) of roleX and roleY can be the same. • Patterns X and Y are said to be disjoint if they have no overlapping roles. X and Y are said to be fully composed if there is a one-to-one mapping between all roles of X and Y.
Page 11 - ... purpose. Grouping of people may be defined by considerations such as sex, color, height, occupation, and income . The concept of indifference as applied to commodity bundles is another example df such equivalence.
Page 3 - Y are equal if and only if they contain the same elements or, equivalently, if and only if every element of X is an element of Y and vice versa.
Page 28 - Prove that every nonempty subset with an upper bound has a least upper bound if and only if every every nonempty subset with a lower bound has a greatest lower bound.
Page 44 - ... 1 Axiom of extensionality If two sets have the same elements then they are identical, Null set axiom There is an empty set, one which contains no elements.
Page 8 - If X is any set, the collection of all subsets of X is a topology on X; it is called the discrete topology.
Page 11 - A partial ordering of a set x is a binary relation on x which is reflexive, antisymmetric, and transitive.
Page 185 - London, 1987. [3] JL Bell, Boolean- Valued Models and Independence Proofs in Set Theory, Oxford University Press, London, 1977.
Page 28 - X and a < b, then there are neighborhoods U of a and V of b such that, whenever xe U and y EV, then x < y.
References to this book
Bibliographic information
| Title | The Joy of Sets: Fundamentals of Contemporary Set Theory Undergraduate Texts in Mathematics |
| Author | Keith Devlin |
| Edition | 2, illustrated |
| Publisher | Springer Science & Business Media, 1994 |
| ISBN | 0387940944, 9780387940946 |
| Length | 194 pages |
| Subjects | › Computers / Data Processing Mathematics / Discrete Mathematics Mathematics / General Mathematics / History & Philosophy Mathematics / Logic |
| Export Citation | BiBTeX EndNote RefMan |