4MATH671FALL2013PROFDAVIDROSSDEPARTMENTOFMATHEMATICSandthereforeacountableunionofsuchintervalsBRCwhereCisthecountablecollectionofsuchopenintervalsDe neA1asaboveWeprovebyinductionthatfora ID: 197978
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2MATH671FALL2013(PROF:DAVIDROSS,DEPARTMENTOFMATHEMATICS)Theorem1.1.(Cantor-Schroeder-Bernstein)IfAhasthesamecardinalityasasubsetofBandviceversa,thencard(A)=card(B).2.OrdinalsJustascardinalityismeanttobeameasureofaset'ssize,ordinalityismeanttobeameasureoftheorderstructureofaset.Whendiscussingaqueueofpeople,forexample,wemightsaythatapersonis5thinline.Thishascardinalityimplications-thesetofpeoplefromthefrontto/includinghimhas5peopleinit-butalsoanordinalityimplication:5indicateshisplaceintheline'sorder.Intuitively,anordinalisawell-orderedsetwhichrepresentsallwellorderedsetswithagiven`ordertype'(thatis,itrepresentsallwell-orderedsetswhichareorder-isomorphictoit).Ordinalsaremeanttogeneralizethenaturalnumbers,andtheorderrelationonordinalswilljustbesetmembership2.Hereisonewaytodenethem:Denition2.1.Anordinalnumberisawell-orderedset( ;)satisfying:8a2 a=fx2 :xagNotethatthismeansthatforanyx;a2 ;x2a()xa,thatis,thewell-orderon isreallyjustsetmembership,andeveryordinalisthesetofitspredecessors.Thisdenitioniscomprehensiveenoughthatfromitonecandeducemanyprop-ertiesofordinals;thisisamajorpartofMath454.Theoneswewillneedare:Lemma2.1.If isanordinal,then:(1)Everya2 isanordinal(2) +1:= [f gisanordinal(3)If andareordinals,then \isanordinal.(4)(Trichotomy)If andareordinalstheneither 2or =or2 (5)If(X;)isawell-orderedset,thenthereisauniqueorder-isomorphismEfromXonto forsomeordinal (6)Iftwoordinalsareorder-isomorphic,thentheyareequal.(7)Everyordinal iseitherasuccessorordinal,thatis, =+1forsome,oritisalimitordinal, =Sf: g.JohnvonNeumanndenedthenaturalnumber0=;tobetherstordinal,andrecursivelyn+1:=f0;1;:::;ng(whichagreeswiththedenitionof\+1"givenintheabovelemma).Inotherwords,theclassofallordinalshasNasitsinitialsegment.Wecandenethingsbyrecursionandprovethingsbyinductiononanyordinalexactlythesamewaywedoonthenaturalnumbers.Forexample,theorder-isomorphismEfromawell-orderedset(X;)toanordinal hasacuterecursivedenition,namely:E(x):=fE(y):yxgThisisprettyterse;youshouldtryitonanimaginarywell-orderedsettoseewhatitmeans.Note,forexample,thatifx0isthesmallestelementofX,thenE(x0)=;. 4MATH671FALL2013(PROF:DAVIDROSS,DEPARTMENTOFMATHEMATICS)(andthereforeacountableunionofsuchintervals),BR=(C)whereCisthecountablecollectionofsuchopenintervals.DeneA!1asabove.Weprovebyinductionthatforall !1;card(A )2@0.card(A0)=@0.Assumecard(A )2@0.EveryelementofA +1iseitheranelementofA ,thecomplementofanelementofA ,orasetdeterminedbyafunctionfromNintoA .Thecardinalityofsuchfunctionsisalso2@0.1ThusA +1istheunionofthreesetseachofcardinalityatmost2@0,soithascardinalityatmost2@0.Finally,A!1istheunionofatmost2@0manysetseachofcardinalityatmost2@0,soittoohascardinalityatmost2@0.Infact,everysingletonsetoftheformfrgforrarealnumberisBorel(why?),soBR=A!1hascardinalityofthecontinuum.We'llseeanotherwaytoprovethislaterinthesemester.Youwon'tndthisfactstated,letaloneproved,inmanyintroanalysisorintroprobabilitytexts,andthatisapity.Bytheway,CantorshowedthatifXisanyset,thencard(X)isstrictlylessthancard(P(X).SincethereareonlyasmanyBorelsetsastherearerealnumbers,thesecardinalityconsiderationsshowthattherearesubsetsofRwhicharenotBorel. 1Youcanshowthisbyndinga1-1functionfrom(thesetoffunctionsfromNtoR)intoR,thenapplyingtheCantor-Schroeder-BernsteinTheorem.