伯克利 (第1/1页)

    AcardinalκisaBerkeleycardinal，ifforanytransitivesetMwithκ∈Mandanyordinalα＜κthereisanelementaryembeddingj:MMwithα＜critj＜κ.ThesecardinalsaredefinedinthecontextofZFsettheorywithouttheaxiomofchoice.

    TheBerkeleycardinalsweredefinedbyW.HughWoodininabout1992athisset-theoryseminarinBerkeley，withJ.D.Hamkins，A.Lewis，D.Seabold，G.HjorthandperhapsR.Solovayintheaudience，amongothers，issuedasachallengetorefuteaseeminglyover-stronglargecardinalaxiom.Nevertheless，theexistenceofthesecardinalsremainsunrefutedinZF.

    IfthereisaBerkeleycardinal，thenthereisaforcingextensionthatforcesthattheleastBerkeleycardinalhascofinalityω.ItseemsthatvariousstrengtheningsoftheBerkeleypropertycanbeobtainedbyimposingconditionsonthecofinalityofκ(Thelargercofinality，thestrongertheoryisbelievedtobe，uptoregularκ).IfκisBerkeleyanda，κ∈MforMtransitive，thenforanyα＜κ，thereisaj:MMwithα＜critj＜κandj(a)=a.

    Acardinalκiscalledproto-BerkeleyifforanytransitiveMκ，thereissomej:MMwithcritj＜κ.Moregenerally，acardinalisα-proto-BerkeleyifandonlyifforanytransitivesetMκ，thereissomej:MMwithα＜critj＜κ，sothatifδ≥κ，δisalsoα-proto-Berkeley.Theleastα-proto-Berkeleycardinaliscalledδα.

    WecallκaclubBerkeleycardinalifκisregularandforallclubsCκandalltransitivesetsMwithκ∈Mthereisj∈E(M)withcrit(j)∈C.

    WecallκalimitclubBerkeleycardinalifitisaclubBerkeleycardinalandalimitofBerkeleycardinals.

    Relations

    IfκistheleastBerkeleycardinal，thenthereisγ＜κsuchthat(Vγ，Vγ 1)ZF2 “ThereisaReinhardtcardinalwitnessedbyjandanω-hugeaboveκω(j)”(Vγ，Vγ 1)ZF2 “ThereisaReinhardtcardinalwitnessedbyjandanω-hugeaboveκω(j)”.

    Foreveryα，δαisBerkeley.ThereforeδαistheleastBerkeleycardinalaboveα.

    Inparticular，theleastproto-Berkeleycardinalδ0isalsotheleastBerkeleycardinal.

    IfκisalimitofBerkeleycardinals，thenκisnotamongtheδα.

    EachclubBerkeleycardinalistotallyReinhardt.

    TherelationbetweenBerkeleycardinalsandclubBerkeleycardinalsisunknown.

    IfκisalimitclubBerkeleycardinal，then(Vκ，Vκ 1)“ThereisaBerkeleycardinalthatissuperReinhardt”.Moreover，theclassofsuchcardinalsarestationary.

    ThestructureofL(Vδ 1)

    IfδisasingularBerkeleycardinal，DC(cf(δ) )，andδisalimitofcardinalsthemselveslimitsofextendiblecardinals，thenthestructureofL(Vδ 1)issimilartothestructureofL(Vλ 1)undertheassumptionλi.e.thereissomej:L(Vλ 1)L(Vλ 1).Forexample，Θ=ΘL(Vδ 1)Vδ 1，thenΘisastronglimitinL(Vδ 1)，δ isregularandmeasurableinL(Vδ 1)，andΘisalimitofmeasurablecardinals.