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1、NON-STANDARDANALYSISABRAHAMROBINSONUniversityofCalifornia,LosAngelesI966NORTH-H0LLANDPUBLISHINGC0MPANYAMSTERDAMNopartofthisbookmaybereproducedinanyformbyprint,microfirmoranyothermeanswithoutwrittenpermissionfromthepublisherPRINTEDINTHENETHERLANDSToRedePREFACEJevoisplusquejamaisqu'ilnefautjugerd
2、eriensursagrandeurapparente.0Dieu!quiavezdonne'uneintelligenceddessubstancesquipamissentsimeprisables,I'infinimentpetitvouscoiiteautantquel'injinimentgrand.VOLTAIRE,MicromegasInthefallof1960itoccurredtomethattheconceptsandmethodsofcontemporaryMathematicalLogicarecapableofprovidingasuitableframe
3、workforthedevelopmentoftheDifferentialandIntegralCalculusbymeansofinfinitelysmallandinfinitelylargenumbers.IfirstreportedmyideasinaseminartalkatPrincetonUniversity(November1960)and,later,inanaddressattheannualmeetingoftheAssociationforSymbolicLogic(January1961)andinapaperpublishedintheProceedin
4、gsoftheRoyalAcademyofSciencesofAmsterdam(ROBINSON[1961]).There-sultingsubjectwascalledbymeNon-standardAnalysissinceitinvolvesandwas,inpart,inspiredbytheso-calledNon-standardmodelsofArithmeticwhoseexistencewasfirstpointedoutbyT.Skolem.Intheinterveningyears,Non-standardAnalysishasdevelopedcon-sid
5、erablyinseveraldirections.Sincemanyoftheresultshavebeenreportedsofaronlyincoursesorlectures,andinmimeographedreports,itwasthoughtthatabookdedicatedentirelytothissubjectwouldbeinorder,Overtheyears,mythinkinginthisareahasbeenenlivenedbydis-cussionswithseveralcolleagues,amongwhomIventuretomentionh
6、ereR.Arens,C.C.Chang,A.Erdelyi,A.Horn,G.Kreisel,I.Lakatos,andJ.B.Rosser.SpecialthanksareduetoW.A.J.Luxemburg,whoselecturesandlecturenotesonNon-standardAnalysishavedonemuchtomakethesubjectknownamongmathematicians.VIlIPREFACEIacknowledgewiththanksthesupportreceivedfromtheNationalScienceFoundation
7、(GrantsGP-1812andGP-4038)inconnectionwithmuchoftheresearchincorporatedinthepresentbook.IalsowishtomentionthatChapterVIisbasedonresearchsponsoredinpartbytheAirForceOfficeofScientificResearchandpublishedpreviouslyinreportform(ROBINS
