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> A cardinal κ is a Berkeley cardinal, if for any transitive set $M$ with $κ∈M$ and any ordinal $α<κ$ there is an elementary embedding $j : M → M$ with $\alpha<\text{crit}(j)<\kappa$.
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My question is about the restrictions involved in that definition of $\kappa \in M$? and of $M$ being *transitive*? and of $M$ being a *set*? why not the following? 


**....if for any *class* $M$ with $κ \subseteq M$ and any ordinal $α<κ$ there is an elementary embedding $j : M → M$ with $\alpha<\text{crit}(j)<\kappa$.**


Where the underlying theory is $MK$ like, with the axiom of limitation of size replaced by an axiom of limitation of size on sets, more specifically any class that is subnumerous to a set, is a set. And of course without $AC$.



Why the restrictions in the definition of Berkeley cardinals?
Dropping transitivity doesn't actually add anything: given $M\supseteq\kappa$, just consider $\hat{M}=$ the Mostowski collapse of $M$. Given $\alpha<\kappa$ and $j:\hat{M}\rightarrow \hat{M}$ nontrivial elementary with $\alpha<crit(j)<\kappa$, $j$ lifts to a nontrivial elementary embedding $\hat{j}$ of $M$ into itself with $\alpha<crit(\hat{j})<\kappa$. So if $\kappa$ is Berkeley in the usual sense, it's Berkeley in the nontransitive sense too. 


Finally, if you drop transitivity *and* replace "$\kappa\subseteq M$" by the original "$\kappa\in M$," then the argument above breaks down but the notion becomes inconsistent: taking $M=\{\kappa\}$ doesn't leave room for *any* nontrivial elementary embeddings, for example, let alone ones with critical points (and we can cook up less-silly examples too).




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I had a stupid moment earlier, where I mixed up ZF and ZFC - it's not immediately obvious to me now that replacing "set" with "class" results in inconsistency. I suspect it does however.


