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Set Theory : London Mathematical Society Student Texts - Andras Hajnal

Set Theory

London Mathematical Society Student Texts


Published: 28th November 1999
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This is a classic introduction to set theory in three parts. The first part gives a general introduction to set theory, suitable for undergraduates; complete proofs are given and no background in logic is required. Exercises are included, and the more difficult ones are supplied with hints. An appendix to the first part gives a more formal foundation to axiomatic set theory, supplementing the intuitive introduction given in the first part. The final part gives an introduction to modern tools of combinatorial set theory. This part contains enough material for a graduate course of one or two semesters. The subjects discussed include stationary sets, delta systems, partition relations, set mappings, measurable and real-valued measurable cardinals. Two sections give an introduction to modern results on exponentiation of singular cardinals, and certain deeper aspects of the topics are developed in advanced problems.

'This is a classic introduction to set theory ...'. L'Enseignement Mathematique 'Give this book fifteen minutes attention - you will find that you must buy it! European Maths Society Journal

Prefacep. vii
Introduction to set theoryp. 1
Introductionp. 3
Notation, conventionsp. 5
Definition of equivalence. The concept of cardinality. The Axiom of Choicep. 11
Countable cardinal, continuum cardinalp. 15
Comparison of cardinalsp. 21
Operations with sets and cardinalsp. 28
Examplesp. 36
Ordered sets. Order types. Ordinalsp. 41
Properties of wellordered sets. Good sets. The ordinal operationp. 54
Transfinite induction and recursion. Some consequences of the Axiom of Choice, the Wellordering Theoremp. 66
Definition of the cardinality operation. Properties of cardinalities. The cofinality operationp. 77
Properties of the power operationp. 93
Hints for solving problems marked with * in Part Ip. 101
An axiomatic development of set theoryp. 107
Introductionp. 109
The Zermelo-Fraenkel axiom system of set theoryp. 111
Definition of concepts; extension of the languagep. 114
A sketch of the development. Metatheoremsp. 117
A sketch of the development. Definitions of simple operations and properties (continued)p. 122
A sketch of the development. Basic theorems, the introduction of [omega] and R (continued)p. 124
The ZFC axiom system. A weakening of the Axiom of Choice. Remarks on the theorems of Sections 2-7p. 128
The role of the Axiom of Regularityp. 130
Proofs of relative consistency. The method of interpretationp. 133
Proofs of relative consistency. The method of modelsp. 138
Topics in combinatorial set theoryp. 143
Stationary setsp. 145
[Delta]-systemsp. 159
Ramsey's Theorem and its generalizations. Partition calculusp. 164
Inaccessible cardinals. Mahlo cardinalsp. 184
Measurable cardinalsp. 190
Real-valued measurable cardinals, saturated idealsp. 203
Weakly compact and Ramsey cardinalsp. 216
Set mappingsp. 228
The square-bracket symbol. Strengthenings of the Ramsey counterexamplesp. 234
Properties of the power operation. Results on the singular cardinal problemp. 243
Powers of singular cardinals. Shelah's Theoremp. 259
Hints for solving problems of Part IIp. 272
Bibliographyp. 295
List of symbolsp. 297
Name indexp. 301
Subject indexp. 303
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780521593441
ISBN-10: 0521593441
Series: London Mathematical Society Student Texts
Audience: Tertiary; University or College
Format: Hardcover
Language: English
Number Of Pages: 328
Published: 28th November 1999
Country of Publication: GB
Dimensions (cm): 22.86 x 15.24  x 2.24
Weight (kg): 0.65