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1、IntroductiontoModernSetTheoryJudithRoitmanDecember6,20111formyfather,wholovedmathematics1RevisedEditionCopyright2011byJudithRoitman,ThisbookislicensedforuseunderaCreativeCommonsLicense(CCBY-NC-ND3.0).Youmaydownload,share,andusethisworkatnocharge,butmaynotmodifynor
2、sellit.1TableofContentsPreface31.Preliminaries51.1Partiallyorderedsets1.2Thestructureofpartiallyorderedsets1.3Equivalencerelations1.4Well-orderedsets1.5Mathematicalinduction1.6Filtersandideals1.7Exercises2.Theoriesandtheirmodels222.1First-orderlanguages2.2Theories
3、2.3Modelsandincompleteness2.4Exercises3.Theaxioms,partI293.1Whyaxioms?3.2Thelanguage,some niteoperations,andtheaxiomofextensionality3.2.1Modelsofextensionality3.3Pairs3.3.1Modelsofpairing3.4Cartesianproducts3.5Union,intersection,separation3.5.1Modelsofunionandsepa
4、ration3.6Natlast3,6,1Modelsofin nity3.7Powersets3.7.1Modelsofpowerset3.8Replacement3.8.1.Modelsofreplacement3.9Exercises24.Regularityandchoice464.1Regularity,partI4,2Transitivesets4.3A rstlookatordinals4.4Regularity,partII4.5Choice4.6EquivalentstoAC4.6.1Modelsofre
5、gularityandchoice4.7Embeddingmathematicsintosettheory4.7.1Z4.7.2Q4.7.3R4.8Exercises5.In nitenumbers625.1Cardinality5.2Cardinalitywithchoice5.3Ordinalarithmetic5.4Cardinalarithmetic5.5Co nality5.6In niteoperationsandmoreexponentiation5.7Counting5.8Exercises6.Twomod
6、elsofsettheory856.1AsetmodelforZFC6.2Theconstructibleuniverse6.3Exercises7.Semi-advancedsettheory937.1Partitioncalculus7.2Trees7.3Measurablecardinals7.4Cardinalinvariantsofthereals37.5CHandMA7.6Stationarysetsand}7.7Exercises4PrefaceWhen,inearlyadolescence,I rstsaw
7、theproofthattherealnumberswereuncountable,Iwashooked.Ididn'tquiteknowonwhat,butItreasuredthatproof,wouldrunitoverinmymind,andwasamazedthattherestoftheworlddidn'tsharemyenthusiasm.Muchlater,learningthatsettheoristscouldactuallyprovesomebasicmathematicalquestionstob
8、eunanswerable,andthatlargein nitenumberscoulde ectthestructureofthereals
9、thenumberlinefamiliartoallofusfromtheearlygrades
10、Iwasevenmoreastonishedthatthew
