For every α，δα is Berkeley. Therefore δα is the least Berkeley cardinal above α.
In particular， the least proto-Berkeley cardinal δ0 is also the least Berkeley cardinal.
If κ is a limit of Berkeley cardinals， then κ is not among the δα.
Each club Berkeley cardinal is totally Reinhardt.
The relation between Berkeley cardinals and club Berkeley cardinals is unknown.
If κ is a limit club Berkeley cardinal， then (Vκ，Vκ+1)?“There is a Berkeley cardinal that is super Reinhardt”. Moreover， the class of such cardinals are stationary.
The structure of L(Vδ+1)
If δ is a singular Berkeley cardinal， DC(cf(δ)+)， and δ is a limit of cardinals themselves limits of extendible cardinals， then the structure of L(Vδ+1) is similar to the structure of L(Vλ+1) under the assumption λ is I0; i.e. there is some j:L(Vλ+1)?L(Vλ+1). For example，Θ=ΘL(Vδ+1)Vδ+1， then Θ is a strong limit in L(Vδ+1)，δ+ is regular and measurable in L(Vδ+1)， and Θ is a limit of measurable cardinals.
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