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.
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