伯克利

    AcardinalκisaBerkeleycardinal，ifforanytransitivesetMwithκ∈Mandanyordinalα＜κthereisanelementaryembeddingj:M≺Mwithα＜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:M≺Mwithα＜critj＜κandj(a)=a

    Acardinalκiscalledproto-BerkeleyifforanytransitiveM∋κ，thereissomej:M≺Mwithcritj＜κ.Moregenerally，acardinalisα-proto-BerkeleyifandonlyifforanytransitivesetM∋κ，thereissomej:M≺Mwithα＜critj＜κ，sothatifδ≥κ，δisalsoα-proto-Berkeley.Theleastα-proto-Berkeleycardinaliscalledδα

    WecallκaclubBerkeleycardinalifκisregularandforallclubsC⊆κandalltransitivesetsMwithκ∈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

    EachclubBerkeleycardinalistotallyReinhard

    TherelationbetweenBerkeleycardinalsandclubBerkeleycardinalsisunknown

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

    ThestructureofL(Vδ+1)

    IfδisasingularBerkeleycardinal，DC(cf(δ)+)，andδisalimitofcardinalsthemselveslimitsofextendiblecardinals，thenthestructureofL(Vδ+1)issimilartothestructureofL(Vλ+1)undertheassumptionλisI0;i.e.thereissomej:L(Vλ+1)≺L(Vλ+1).Forexample，Θ=ΘL(Vδ+1)Vδ+1，thenΘisastronglimitinL(Vδ+1)，δ+isregularandmeasurableinL(Vδ+1)，andΘisalimitofmeasurablecardinals