acardinalkisaberkeleycardinal，ifforanytransitivesetmwithk∈mandanyordinalα＜kthereisanelementaryembeddingj:mmwithα＜critj＜k.thesecardinalsaredefinedinthecontextofzfsettheorywithouttheaxiomofchoice.

    theberkeleycardinalsweredefinedbyw.hughwoodininabout1992athisset-theoryseminarinberkeley，withj.d.hamkins，a.lewis，d.seabold，g.hjorthandperhapsr.solovayintheaudience，amongothers，issuedasachallengetorefuteaseeminglyover-stronglargecardinalaxiom.nevertheless，theexistenceofthesecardinalsremainsunrefutedinzf.

    ifthereisaberkeleycardinal，thenthereisaforcingextensionthatforcesthattheleastberkeleycardinalhascofinalityw.itseemsthatvariousstrengtheningsoftheberkeleypropertycanbeobtainedbyimposingconditionsonthecofinalityofk(thelargercofinality，thestrongertheoryisbelievedtobe，uptoregulark).ifkisberkeleyanda，k∈mformtransitive，thenforanyα＜k，thereisaj:mmwithα＜critj＜kandj(a)=a.

    acardinalkiscalledproto-berkeleyifforanytransitivemk，thereissomej:mmwithcritj＜k.moregenerally，acardinalisα-proto-berkeleyifandonlyifforanytransitivesetmk，thereissomej:mmwithα＜critj＜k，sothatifδ≥k，δisalsoα-proto-berkeley.theleastα-proto-berkeleycardinaliscalledδα.

    wecallkaclubberkeleycardinalifkisregularandforallclubsckandalltransitivesetsmwithk∈mthereisj∈e(m)withcrit(j)∈c.

    wecallkalimitclubberkeleycardinalifitisaclubberkeleycardinalandalimitofberkeleycardinals.

    relations

    ifkistheleastberkeleycardinal，thenthereisγ＜ksuchthat(vγ，vγ+1)zf2+“thereisareinhardtcardinalwitnessedbyjandanw-hugeabovekw(j)”(vγ，vγ+1)zf2+“thereisareinhardtcardinalwitnessedbyjandanw-hugeabovekw(j)”.

    foreveryα，δαisberkeley.thereforeδαistheleastberkeleycardinalaboveα.

    inparticular，theleastproto-berkeleycardinalδ0isalsotheleastberkeleycardinal.

    ifkisalimitofberkeleycardinals，thenkisnotamongtheδα.

    eachclubberkeleycardinalistotallyreinhardt.

    therelationbetweenberkeleycardinalsandclubberkeleycardinalsisunknown.

    ifkisalimitclubberkeleycardinal，then(vk，vk+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.