non cohen oracle c.c.c

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1、NONCOHENORACLEC.C.C.SH669SaharonShelahInstituteofMathematicsTheHebrewUniversityJerusalem,IsraelRutgersUniversityMathematicsDepartmentNewBrunswick,NJUSAAbstract.Theoraclec.c.c.iscloselyrelatedtoCohenforcing.Duringaniterationwecan“omitatype”;i.e.preserve“theintersectionofagivenfamilyofBorelsetsof

2、realsisempty”providedthatCohenforcingsatisfiesit.Wegeneralizethistoothercases.In§1wereplaceCohenby“nicely”definablec.c.c.,dotheparalleloftheoraclec.c.c.andendwithacriterionforextractingasubforcing(notacompletesubforcing,⋖!)ofagivennicelyoneandsatisfyingtheoracle.arXiv:math/0303294v1[math.LO]24Mar

3、2003IwouldliketothankAliceLeonhardtforthebeautifultyping.FirstTyped-1/9/98Latestrevision-02/Nov/5TypesetbyAMS-TEX12SAHARONSHELAH§0IntroductionThisanswersaquestionfom[Sh:b,Ch.IV](thechapterdealingwiththeoraclec.c.c.)askingtoreplaceCohenbye.g.random.Laterwewilldealwiththeparallelfororacleproperan

4、dforthecase¯ϕαisa(definitionofa)nepforcing.AnapplicationwillappearinaworkwithT.Bartoszynski.Howdoweusethisframework?Westartwithauniversesatisfying♦ℵ1andprobably2ℵ1=ℵandchoosehS∗:i<ωi,S∗⊆S∗⊆ωsuchthatS∗/Dis2i2i1iω1∗∗strictlyincreasingand♦S∗S∗andforsimplicitySi⊆Si+1whereDω1istheclubi+1ifilteronω1.W

5、echoosebyinductiononi<ω2,ac.c.c.forcingPiofcardinalityℵ1asequenceM¯i=hMi:α∈S∗ianda1-commitment.αiTheyareincreasingintherelevantsenseandtheworkatlimitstagesisdonebythegeneralclaimshere.Fori=j+1wehavesomefreedominchoosingPi+1,usuallyP=P∗Q.So,workinginVPi,Qhastosatisfya0-commitmentoni+1iii∼S∗,andw

6、elikeittosatisfysometask,possiblyconnectedwithsomeX⊆RV[Pi],iisayX=X[G]towardwhatevertaskwehave.WeessentiallyhavetochooseM¯iiiPi∼suchthatM¯i↾S∗=M¯jbutwehavefreedomtochoosehM¯i:α∈S∗S∗iandajαij0-commitmentonS∗S∗.TherealsgenericforthechosenforcingnotionaswellasijMiforα∈S∗S∗canbechosenconsidering

7、X.E.g.MicanbetheMostowskiαijiαCollapseofsomeM≺(H(ℵ),∈)towhichP,M¯j,xandXbelongs.2jjj∼Soreallythiscorrespondstotheomittingtypeasin[Sh:e,XI].NONCOHENORACLEC.C.C.SH6693§1Non-Cohenoraclec.c.c.1.1Hypothesis.∗(a)weassumeCH,moreover♦S∗wh