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1、WolframPohlersProofTheoryTheFirstStepintoImpredicativityABCWolframPohlersUniversitatM¨unster¨Inst.MathematischeLogikundGrundlagenforschungEinsteinstr.6248149M¨unsterGermanyISBN978-3-540-69318-5e-ISBN978-3-540-69319-2LibraryofCongressControlNumber:2008930149MathematicsSubjectClassificati
2、on(2000):03F03,03F05,03F15,03F25,03F30,03F35°c2009Springer-VerlagBerlinHeidelbergThisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broad-casting,reproductiononmicr
3、ofilmorinanyotherway,andstorageindatabanks.DuplicationofthispublicationorpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9,1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.ViolationsareliabletoprosecutionundertheGermanCopyrightL
4、aw.Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.Coverdesign:WMXDesignGmbH,HeidelbergPrintedonacid-freepaper
5、987654321springer.comToRenatePrefaceThekernelofthisbookconsistsofaseriesoflecturesoninfinitaryprooftheorywhichIgaveduringmytimeattheWestfalischeWilhelms–Universit¨atinM¨unster.¨ItwasplannedasasuccessorofSpringerLectureNotesinMathematics1407.How-ever,whenpreparingit,Idecidedtoalsoinclude
6、materialwhichhasnotbeentreatedinSLN1407.SincetheappearanceofSLN1407manyinnovationsintheareaofor-dinalanalysishavetakenplace.Justtomentionthoseofthemwhichareaddressedinthisbook:Buchholzsimplifiedlocalpredicativitybytheinventionofoperatorcontrolledderivations(cf.Chapter9,Chapter11);Weierm
7、anndetectedapplicationsofmethodsofimpredicativeprooftheorytothecharacterizationoftheprovablerecursivefunctionsofpredicativetheories(cf.Chapter10);BeckmannimprovedGentzen’sboundednesstheorem(whichappearsasStageTheorem(Theorem6.6.1)inthisbook)toTheorem6.6.9,atheoremwhichisverysatisfyin
