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1、LectureNotesinMathematicsEditedbyA.DoldandB.Eckmann617KeithJ.DevlinTheAxiomofConstructibility:AGuidefortheMathematicianIIISpringer-VerlagBerlinHeidelbergNewYork1977AuthorKeithJ.DevlinDepartmentofMathematicsUniversityofLancasterLancaster/EnglandLibraryofCongre
2、ssCataloginginPublicationDataDevli%KeithJTheaxiomofconstructibility.(Lecturenotesinmathematics;617)Bibliography:p.Includesindex.i.Axiomofeonstruetibility.I.Title.II-Se-ries:Lecturenotesinmathematics(Berlin)~617.QA3oL°8no,617[QA248]510'.8s[511'.3]77-17119AMSSu
3、bjectClassifications(1970):02K15,02K25,02K05,04-01,04A30,20A10,20K35,54D15ISBN3-540-08520-3Springer-VerlagBerlinHeidelbergNewYorkISBN0-38?-08520-3Springer-VerlagNewYorkHeidelbergBerlinThisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofth
4、ematerialisconcerned,specificallythoseoftranslation,re-printing,re-useofillustrations,broadcasting,reproductionbyphotocopyingmachineorsimilarmeans,andstorageindatabanks.Under§54oftheGermanCopyrightLawwherecopiesaremadeforotherthanprivateuse,afeeispayabletothe
5、publisher,theamountofthefeetobedeterminedbyagreementwiththepublisher.©bySpringer-VerlagBerlinHeidelberg1977PrintedinGermanyPrintingandbinding:BeltzOffsetdruck,Hemsbach/Bergstr.2140/3140-543210PREFACEConsiderthefollowingfourtheoremsofpuremathematics.The~h--Ban
6、achTheoremof/~nalysis:IfFisaboundedlinearfunctionaldefinedonasubspaeeMofaBanachspaceB,thereisanextensionofFtoalinearfunctionalGonBsuchthatllG~l=llF~l.TheNielsen-SchreierTheoremofGroupTheory:IfGisafreegroupandisasubgroupofG,thenHisafreegroup.TheTychonoffProduc
7、tTheoremofGeneralTopology~Theproductofanyfamilyofcompacttopologicalspacesiscompact.TheZermelol~ell-0rderingTheoremofSetTheory:~Verysetcanbewell-ordered.Theabovetheoremshavetwothingsincommon.Firstlytheyareallfundamentalresultsincontemporarymathematics.Secondly
8、,noneofthemcanbeprovedwithouttheaidofsomepowerfulsettheoretic~lass~a~ption:inthiscasethe~xiomofChoice.Now,thereisnothingwrongaboutassumingtheAxiomofChoice.Butletusbesureaboutonething:wear
