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1、BasicSetTheoryDanielMurfetMay16,2006ThroughoutweworkwiththefoundationprovidedbystandardZFC(FCT,Section3).Inparticularwedonotassumeweareworkinginsideanyfixedgrothendieckuniverse.Itseemstomethatsomebasicproofsaboutordinalsinstandardreferencesareflawed,sosincetheendresultisthesameweadoptaslightlydiffere
2、ntdefinitiontomakelifeeasier.Contents1OrdinalNumbers12TransfinitePain43CardinalNumbers74CardinalOperations105RegularCardinals111OrdinalNumbersDefinition1.Givenasetx,weletx+denotethesetx∪{x}.Definition2.Asetxistransitiveifwhenevery∈xwehavey⊆x.Asetxisurtransitiveifitistransitive,andifinadditionwhenevera
3、propersubsety⊂xistransitivewehavey∈x.Wesayxisanordinalifitisurtransitive,andifeveryelementofxisurtransitive.Wetendtouselowercasegreeklettersα,β,γ,...torepresentordinals.Givenordinalsα,βwewriteα≺βforα∈β.Onechecksthatthisisatransitiveirreflexiverelationonordinals.Wewriteαβifα≺βorα=β,andthisdefinesapa
4、rtialorderonordinals.Remark1.Thefollowingobservationsareimmediate•Theemptysetisanordinal.Theemptysetisamemberofanynonemptyordinal.•Anyelementofanordinalisanordinal.•Ifα,βareordinalsthenα≺βifandonlyifα⊂β.Lemma1.Ifαisanordinal,thensoisα+.Proof.Thesetα+isclearlytransitive.Toseethatitisurtransitive,le
5、ty⊂α∪{α}beatransitivepropersubset.Wewanttoshowthaty∈α∪{α}.Supposethaty6=α.Firstweclaimthatα/∈y.Forifthiswerethecase,transitivityofyimpliesα⊂y.Supposet∈yα.Thent∈y⊂α∪{α}sowemusthavet=α.Buttheny⊇α∪{α},whichisacontradiction.Thisshowsthatα/∈y.Butthenymustbeapropersubsetofα,andsinceαisanordinalthismean
6、sy∈α,asrequired.Thisshowsthatα+isurtransitive,anditisclearthateveryelementofα+isurtransitive,soα+isanordinal.1Lemma2.Ifasetcontainsanordinal,thenitcontainsaminimalordinal.Proof.LetXbeasetandsupposeα∈Xforsomeordinalα.ThenthesetZofelementsofXwhichareordinalsisnonempty,andapplyingtheAxiomofFoundation
7、tothissetweobtainanordinalβ∈XwiththepropertythatnoordinalinβisanelementofX.Proposition3(Minimalelement).LetB(x)beawfwithxfree,andsupposethereexistsanordinalαwithB(α).Thenthereaminimalsuchordinal.Thatis,thereexist
