An Introduction to Nonstandard Real

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1、AnIntroductiontoNonstandardRealAnalysisThisisavolumeinPUREANDAPPLIEDMATHEMATICSASeriesofMonographsandTextbooksEditors:SAMUELEILENBERGANDHYMANBASSAlistofrecenttitlesinthisseriesappearsattheendofthisvolume.AnIntroductiontoNonstandardRealAnalysisALBERTE.HURDDepartmentofMathematicsUniversityofVict

2、oriaVictoria.BritishColumbiaCanadaPETERA.LOEBDepartmentofMathematicsUniversityofIllinoisUrbana,Illinois1985ACADEMICPRESS,INC.(HarcourtBraceJovannvich,Publishers)OrlandoSanDiegoNewYorkLondonTorontoMontrealSydneyTokyoCOPYRIGHTo1985BYACADEMICPRESS,INC.ALLRIGHTSRESERVED.NOPARTOFTHISPUBLICATIONMAYB

3、EREPRODUCEDORTRANSMITTEDINANYFORMORBYANYMEANS.ELECTRONICORMECHANICAL,INCLUDINGPHOTOCOPY.RECORDING.ORANYINFORMATIONSTORAGEANDRETRIEVALSYSTEM.WITHOUTPERMISSIONINWRITINGFROMTHEPUBLISHER.ACADEMICPRESS,INC.Orlando,Florida32887UnitedKingdomEditionpublishedbyACADEMICPRESSINC.(LONDON)LTD.24/28OvalRoad

4、,LondonNWI.7DXLibraryofCongressCataloginginPublicationDataMainentryundertitle:Anintroductiontononstandardrealanalysis.Includesbibliographicalreferencesandindex.1.Mathematicalanalysis,Nonstandard.I.Hurd.A.E.(AlbertEmerson),DATE.11.Loeb,P.A.PA299.82.158198551584-24563SBN0-12-362440-1(alk.paper)P

5、RINTEDINTHEUNITEDSTATkSOFAMtRlCA85868788987654321DedicatedtothememoryofABRAHAMROBINSONThisPageIntentionallyLeftBlankContentsPreface..................................................................................ixChapterIlnfinitesimalsandTheCalculusI.ITheHyperrealNumberSystemasanUltrapower..

6、.................................21.2*-TransformsofRelations...............................................................81.3SimpleLanguagesforRelationalSystems1.4InterpretationofSimpleSentences....................1.5TheTransferPrincipleforSimpleSenten1.6InfiniteNumbers,Infinitesimals,andthe

7、1.7TheHyperintegers..........................1.8SequencesandSeries...................................................................321.9TopologyontheReals1.101.11..................511.12RiemannIntegration..............................