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Set Theory An Introduction To Independence Proofs : Volume 102 - Kenneth Kunen

Set Theory An Introduction To Independence Proofs

Volume 102

Hardcover Published: 15th December 1983
ISBN: 9780444868398
Number Of Pages: 330

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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.

Prefacep. vii
Contentsp. viii
Introductionp. xi
Consistency resultsp. xi
Prerequisitesp. xii
Outlinep. xii
How to use this bookp. xiii
What has been omittedp. xiv
On referencesp. xiv
The axiomsp. xv
The foundations of set theoryp. 1
Why axioms?p. 1
Why formal logic?p. 2
The philosophy of mathematicsp. 6
What we are describingp. 8
Extensionality and Comprehensionp. 10
Relations, functions, and well-orderingp. 12
Ordinalsp. 16
Remarks on defined notionsp. 22
Classes and recursionp. 23
Cardinalsp. 27
The real numbersp. 35
Appendix 1: Other set theoriesp. 35
Appendix 2: Eliminating defined notionsp. 36
Appendix 3: Formalizing the metatheoryp. 38
Exercises for Chapter Ip. 42
Infinitary combinatoricsp. 47
Almost disjoint and quasi-disjoint setsp. 47
Martin's Axiomp. 51
Equivalents of MAp. 62
The Suslin problemp. 66
Treesp. 68
The c.u.b. filterp. 76
[characters not reproducible] and [characters not reproducible]p. 80
Exercises for Chapter IIp. 86
The well-founded setsp. 94
Introductionp. 94
Properties of the well-founded setsp. 95
Well-founded relationsp. 98
The Axiom of Foundationp. 100
Induction and recursion on well-founded relationsp. 102
Exercises for Chapter IIIp. 107
Easy consistency proofsp. 110
Three informal proofsp. 110
Relativizationp. 112
Absolutenessp. 117
The last word on Foundationp. 124
More absolutenessp. 125
The H([kappa])p. 130
Reflection theoremsp. 133
Appendix 1: More on relativizationp. 141
Appendix 2: Model theory in the metatheoryp. 142
Appendix 3: Model theory in the formal theoryp. 143
Exercises for Chapter IVp. 146
Defining definabilityp. 152
Formalizing definabilityp. 153
Ordinal definable setsp. 157
Exercises for Chapter Vp. 163
The constructible setsp. 165
Basic properties of Lp. 165
ZF in Lp. 169
The Axiom of Constructibilityp. 170
AC and GCH in Lp. 173
[characters not reproducible] and [characters not reproducible] in Lp. 177
Exercises for Chapter VIp. 180
Forcingp. 184
General remarksp. 184
Generic extensionsp. 186
Forcingp. 192
ZFC in M[G]p. 201
Forcing with finite partial functionsp. 204
Forcing with partial functions of larger cardinalityp. 211
Embeddings, isomorphisms, and Boolean-valued modelsp. 217
Further resultsp. 226
Appendix: Other approaches and historical remarksp. 232
Exercises for Chapter VIIp. 237
Iterated forcingp. 251
Productsp. 252
More on the Cohen modelp. 255
The independence of Kurepa's Hypothesisp. 259
Easton forcingp. 262
General iterated forcingp. 268
The consistency of MA + [not sign]CHp. 278
Countable iterationsp. 281
Exercises for Chapter VIIIp. 287
Bibliographyp. 305
Index of special symbolsp. 309
General Indexp. 311
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9780444868398
ISBN-10: 0444868399
Series: Studies in Logic and the Foundations of Mathematics
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 330
Published: 15th December 1983
Publisher: Elsevier Science & Technology
Country of Publication: US
Dimensions (cm): 21.9 x 14.9  x 2.4
Weight (kg): 0.55
Edition Number: 8
Edition Type: New edition

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