伯克利
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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.