Simple Theories and Hyperimaginaries

184 Pages·2011·0.972 MB·English

by  Enrique Casanovas

SimpleTheoriesandHyperimaginaries
Inthe1990sByunghanKimandAnandPillaygeneralizedstability,amajormodel-
theoreticideadevelopedbySaharonShelahtwenty-fiveyearsearlier,tothestudy
ofsimpletheories.Thisbookisanup-to-dateintroductiontosimpletheoriesand
hyperimaginaries, with special attention to Lascar strong types and elimination
ofhyperimaginaryproblems.Assumingonlyknowledgeofgeneralmodeltheory,
thefoundationsofforking,stability,andsimplicityarepresentedinfulldetail.The
treatmentofthetopicsisasgeneralaspossible,workingwithstableformulasand
typesandassumingstabilityofthetheoryonlywhennecessary.Theauthoroffers
an introduction to independence relations as well as a full account of canonical
bases of types in stable and simple theories. In the last chapters, the notions of
internality and analyzability are discussed and used to provide a self-contained
proofofeliminationofhyperimaginariesinsupersimpletheories.
Enrique Casanovas is a Professor of Logic and Philosophy of Science in the
Department of Logic, History and Philosophy of Science at the University of
Barcelona.

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Simple Theories and Hyperimaginaries
ENRIQUE CASANOVAS
UniversidaddeBarcelona
association for symbolic logic
cambridgeuniversitypress
Cambridge,NewYork,Melbourne,Madrid,CapeTown,
Singapore,Sa˜oPaulo,Delhi,Tokyo,MexicoCity
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(cid:2)C AssociationofSymbolicLogic2011
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Firstpublished2011
PrintedintheUnitedStatesofAmerica
AcatalogrecordforthispublicationisavailablefromtheBritishLibrary.
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Casanovas,Enrique,1957–
Simpletheoriesandhyperimaginaries/EnriqueCasanovas.
p. cm.–(Lecturenotesinlogic)
Includesbibliographicalreferencesandindex.
ISBN978-0-521-11955-9
1.Modeltheory. 2.First-orderlogic. 3.Hyperspace. I.Title. II.Series.
QA9.7.C372011
511.3(cid:3)4–dc22 2011005729
ISBN978-0-521-11955-9Hardback
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CONTENTS
Preface.............................................................. xi
Chapter1. Preliminaries.......................................... 1
Chapter2. ϕ-types,stabilityandsimplicity....................... 11
Chapter3. ∆-typesandthelocalrankD((cid:240),∆,k)................. 19
Chapter4. Forking............................................... 25
Chapter5. Independence.......................................... 31
Chapter6. ThelocalrankCB ((cid:240))................................ 37
∆
Chapter7. Heirsandcoheirs...................................... 43
Chapter8. Stableforking......................................... 47
Chapter9. Lascarstrongtypes................................... 53
Chapter10. Theindependencetheorem............................ 63
Chapter11. Canonicalbases...................................... 69
Chapter12. Abstractindependencerelations..................... 75
Chapter13. Supersimpletheories.................................. 85
Chapter14. Moreranks........................................... 93
Chapter15. Hyperimaginaries.....................................101
Chapter16. Hyperimaginaryforking..............................109
Chapter17. Canonicalbasesrevisited.............................121
Chapter18. Eliminationofhyperimaginaries......................131
ix