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1、ELEMENTSOFMATHEMATICALLOGIC(MODELTHEORY)G.KREISELJ.L.KRIVINEStanfordUni~ers~~yUni~ers~t~deParis1967NORTH-HOLLANDPUBLISHINGCOMPANYAMSTERDAM8North-HollandPublishingCompany-Amsterdam-1967Allrightsreserved.Nopartofthisbookmaybereproducedinanyformbyprint,photoprint,micr
2、ofilmoranyothermeanswithoutwrittenpermissionfromthepublisherLibraryofCongressCatalogCardNumber:67-20006.PRINTEDINTHENETHERLANDSPREFACEThisbookpresentstheprinciplesoftheAxiomaticMethod,hereformulatedinsettheoretic,alsocalled:semantic,terms.Thebasicnotionsinvolvedare
3、:differentkindsoflanguages;theirrealizations(typesofmathematicalstructures);andmodels(ofaformulainthelanguageconsidered,i.e.,therealizationsofthelanguagewhichsatisfytheformula).Fromthemarederivedthenotionsofconsequence(aconclusionAbeingaconsequenceof“axioms”d,formu
4、latedinalanguage9,ifeveryrealizationof9whichsatisfieseachformulaofdalsosatisfiesA)andofdejinability(inarealizationof9bymeansofaformulaof9).Consequenceanddefinabilityarethetwomaintopicsherestudied.ThemostgeneralresultsontheAxiomaticMethodknown,applytoaxiomaticsystem
5、sformulatedinthelanguageofpredicatelogicofjirstorderrestrictedtofiniteformulas.Muchofthistheorycanbegeneralizedtosuitableinfiniteformulasoffirstorder,lesstolanguagesofhigherorder,evenwhentheyarerestrictedtofiniteformulas.Thelastchaptercontainssomeinformationonsuchg
6、eneralizations.Thetreatmentissettheoreticinthatthebasicnotionsabovearede-finedinthevocabularyofcurrentsettheory:sets,membershiprelation,logicaloperations.Thisbookcontainstheelementary,moreorlessclassical,resultsofitssubject.Eachofitseightchaptersisprecededbyasummar
7、ywhichnotonlyindicatesthegeneralcontentofthechapter,butalsotherelationoftheexercisestothemaintheorems.AppendixIgivesanideaofthekindofapplicationstocurrentmathe-maticsthatcanbeexpectedfromageneraltheoryoftheAxiomaticMethod.Knowledgeofthemaintextisnotassumed.Appendix
8、I1isintendedforreaderswithsomeinterest(andback-ground!)inthephilosophyofmathematics.PartsAandBsketchtheso-calledsemanticandsyntactic(better:setth
