抽象置数理论 Abstract Set Theory

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1、NOTREDAMEMATHEMATICALLECTURESNumber8ABSTRACTSETTHEORYbyTHORALFA.SKOLEMProfessorofMathematicsUniversityofOslo,NorwayNOTREDAME,INDIANA1962Copyright1962UNIVERSITYOFNOTREDAMECOMPOSITIONBYBELJAN,ANNARBOR,MICHIGANPHOTOLITHOPRINTEDBYGUSHING-MALLOY,INC.ANNARBOR,MICHIGAN,UNITEDSTATESOFAMERICA19

2、62PREFACEThefollowingpagescontainaseriesoflecturesonabstractsettheorygivenattheUniversityofNotreDameduringtheFallSemester1957-58.AftersomehistoricalremarksthechiefideasofCantor'stheory,nowusuallycalledthenaivesettheory,areexplained.ThentheaxiomatictheoryofZermelo-Fraenkelisde-velopedan

3、dsomecriticalremarksadded.Inparticulartheset-theoreticrelativismisemphasizedasanaturalconsequenceoftheapplicationofLowenheim'sTheoremontheaxiomsofsettheory.Otherversionsofaxiomaticsettheorywhichlogicallyareofverysimilarcharacterarenotdealtwith.However,thesimpletheoryoftypes,Quine'stheo

4、ryandtheramifiedtheoryoftypesaretreatedtoacertainextent.AlsoLorenzen'soperativemathematicsandtheintuitionistmathematicsareoutlined.Further,thereisashortremarkonthepossibilityoffinitistmathematicsinastrictsenseandfinallysomehintsaregivenaboutthepossibilityofasettheorybasedonalogicwithan

5、in-finitenumberoftruthvalues.Thebook"TransfiniteZahlen"byH.Bachmannhasbeenveryusefulinparticularforthewritingofparts6and8.Somereferencestotheliteratureonthesesubjectsoccurscatteredinthetext,butnoattempthasbeenmadetosetupacompletelist.Suchataskseemsindeedscarcelyworthwhile,becauseveryex

6、tensiveandcompletelistscanbefoundbothinthementionedbookofBachmannandinthebook"AbstractSetTheory"byA.Fraenkel.Th.Skolem.CONTENTS1.Historicalremarks.OutlinesofCantor'stheory12.Orderedsets.AtheoremofHausdorff73.Axiomaticsettheory.AxiomsofZermeloandFraenkel124.Thewell-orderingtheorem195.Or

7、dinalsandalephs226.Someremarksonfunctionsofordinalnumbers287.Ontheexponentiationofalephs328.Setsrepresentingordinals359.Thenotions"finite"and"infinite"3810.Thesimpleinfinitesequence.Developmentofarithmetic4111.Someremarksonthenatureoftheset-theoreticaxioms.Theset-theoreticrelativism4