Set Theory

An Introduction to Independence Proofs

By: Kenneth Kunen

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Hardcover | 1 December 1983 | Edition Number 8

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Hardcover
332 Pages

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New edition

Dimensions(cm)
21.59 x 13.97 x 1.91

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Studies in Logic and the Foundations of Mathematics, Volume 102: Set Theory: An Introduction to Independence Proofs offers an introduction to relative consistency proofs in axiomatic set theory, including combinatorics, sets, trees, and forcing.

The book first tackles the foundations of set theory and infinitary combinatorics. Discussions focus on the Suslin problem, Martin's axiom, almost disjoint and quasi-disjoint sets, trees, extensionality and comprehension, relations, functions, and well-ordering, ordinals, cardinals, and real numbers. The manuscript then ponders on well-founded sets and easy consistency proofs, including relativization, absoluteness, reflection theorems, properties of well-founded sets, and induction and recursion on well-founded relations. The publication examines constructible sets, forcing, and iterated forcing. Topics include Easton forcing, general iterated forcing, Cohen model, forcing with partial functions of larger cardinality, forcing with finite partial functions, and general extensions.

The manuscript is a dependable source of information for mathematicians and researchers interested in set theory.

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Preface	p. vii
Contents	p. viii
Introduction	p. xi
Consistency results	p. xi
Prerequisites	p. xii
Outline	p. xii
How to use this book	p. xiii
What has been omitted	p. xiv
On references	p. xiv
The axioms	p. xv
The foundations of set theory	p. 1
Why axioms?	p. 1
Why formal logic?	p. 2
The philosophy of mathematics	p. 6
What we are describing	p. 8
Extensionality and Comprehension	p. 10
Relations, functions, and well-ordering	p. 12
Ordinals	p. 16
Remarks on defined notions	p. 22
Classes and recursion	p. 23
Cardinals	p. 27
The real numbers	p. 35
Appendix 1: Other set theories	p. 35
Appendix 2: Eliminating defined notions	p. 36
Appendix 3: Formalizing the metatheory	p. 38
Exercises for Chapter I	p. 42
Infinitary combinatorics	p. 47
Almost disjoint and quasi-disjoint sets	p. 47
Martin's Axiom	p. 51
Equivalents of MA	p. 62
The Suslin problem	p. 66
Trees	p. 68
The c.u.b. filter	p. 76
[characters not reproducible] and [characters not reproducible]	p. 80
Exercises for Chapter II	p. 86
The well-founded sets	p. 94
Introduction	p. 94
Properties of the well-founded sets	p. 95
Well-founded relations	p. 98
The Axiom of Foundation	p. 100
Induction and recursion on well-founded relations	p. 102
Exercises for Chapter III	p. 107
Easy consistency proofs	p. 110
Three informal proofs	p. 110
Relativization	p. 112
Absoluteness	p. 117
The last word on Foundation	p. 124
More absoluteness	p. 125
The H([kappa])	p. 130
Reflection theorems	p. 133
Appendix 1: More on relativization	p. 141
Appendix 2: Model theory in the metatheory	p. 142
Appendix 3: Model theory in the formal theory	p. 143
Exercises for Chapter IV	p. 146
Defining definability	p. 152
Formalizing definability	p. 153
Ordinal definable sets	p. 157
Exercises for Chapter V	p. 163
The constructible sets	p. 165
Basic properties of L	p. 165
ZF in L	p. 169
The Axiom of Constructibility	p. 170
AC and GCH in L	p. 173
[characters not reproducible] and [characters not reproducible] in L	p. 177
Exercises for Chapter VI	p. 180
Forcing	p. 184
General remarks	p. 184
Generic extensions	p. 186
Forcing	p. 192
ZFC in M[G]	p. 201
Forcing with finite partial functions	p. 204
Forcing with partial functions of larger cardinality	p. 211
Embeddings, isomorphisms, and Boolean-valued models	p. 217
Further results	p. 226
Appendix: Other approaches and historical remarks	p. 232
Exercises for Chapter VII	p. 237
Iterated forcing	p. 251
Products	p. 252
More on the Cohen model	p. 255
The independence of Kurepa's Hypothesis	p. 259
Easton forcing	p. 262
General iterated forcing	p. 268
The consistency of MA + [not sign]CH	p. 278
Countable iterations	p. 281
Exercises for Chapter VIII	p. 287
Bibliography	p. 305
Index of special symbols	p. 309
General Index	p. 311
Table of Contents provided by Ingram. All Rights Reserved.

Set Theory

An Introduction to Independence Proofs

At a Glance

Hardcover

Shipping

How to return your order

Defective items

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Other Editions and Formats

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