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1、THEAXIOMOFCHOICETHOMASJ.JECHStateUniversityofNewYorkatBufaloandTheInstituteforAdvancedStudyPrinceton,NewJersey1973NORTH-HOLLANDPUBLISHINGCOMPANY-AMSTERDAMLONDONAMERICANELSEVIERPUBLISHINGCOMPANY,INC.-NEWYORK0NORTH-HOLLANDPUBLISHINGCOMPANY-1973ANRightsReserved.Nopartofthispublicationmaybereproduced,st
2、oredinaretrievalsystemortransmitted,inanyformorbyanymeans,electronic,mechanical,photocopying,recordingorotherwise,withoutthepriorpermissionoftheCopyrightowner.LibraryofCongressCatalogCardNumber73-15535North-HollandISBNfortheseries0720422000forthisvolume0120422152AmericanElsevierISBN0444104844Publish
3、edby:North-HollandPublishingCompany-AmsterdamNorth-HollandPublishingCompany,Ltd.-LondonSoledistributorsfortheU.S.A.andCanada:AmericanElsevierPublishingCompany,Inc.52VanderbiltAvenueNewYork,N.Y.10017PRINTEDINTHENETHERLANDSTomyparentsPREFACEThebookwaswritteninthelongBuffalowinterof1971-72.Itisanattemp
4、ttoshowtheplaceoftheAxiomofChoiceincontemporarymathe-matics.MostofthematerialcoveredinthebookdealswithindependenceandrelativestrengthofvariousweakerversionsandconsequencesoftheAxiomofChoice.AlsoincludedaresomeotherresultsthatIfoundrelevanttothesubject.Theselectionofthetopicsandresultsisfairlycompreh
5、ensive,neverthelessitisaselectionandassuchreflectsthepersonaltasteoftheauthor.Sodoesthetreatmentofthesubject.ThemaintoolusedthroughoutthetextisCohen’smethodofforcing.Itriedtouseasmuchsimilaritybetweensym-metricmodelsofZFandpermutationmodelsofZFAaspossible.Therelationbetweenthesetwomethodsisdescribed
6、inChapter6,whereIpre-senttheEmbeddingTheorem(Jech-Sochor)andtheSupportTheorem(Pincus).ConsistencyandindependenceoftheAxiomofChoice(duetoGodelandCohen,respectively)arepresentedinChapters3and5.Chapter4introducesthepermutationmodels(Fraenkel,Mostowski,Specker).Chapters2and10aredevotedtotheuseoftheAxiom
7、ofChoiceinmathematics.Chapter2containsexamplesofproofsusingtheAxiomofChoice,whereasChapter10providescounterexamplesintheabsenceoftheAxiomofChoice.Chapters7and8dealwithvariousconsequencesoftheAxiomofCh
