# Why the restrictions in the definition of Berkeley cardinals?

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 $$\alpha<\text{crit}(j)<\kappa$$.

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 $$AC$$.

• I think assuming the condition for every class doesn't immediately lead to a contradiction, but it might increase the strength of the notion - suppose there is an inaccessible $\lambda$ above a Berkeley $\kappa$. Consider $V_\lambda$ - every class in it is a set in $V$, and an elementary embedding from it to itself is again a class in $V_\lambda$, so $V_\lambda$ satisfies the Berkeley condition for all classes. One reason for restricting the definition to sets is that it turns the notion into something expressible in ZF - you can't quantify, not even explicitly talk about, classes there. – Wojowu Jan 2 '19 at 19:25

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 $$\alpha, $$j$$ lifts to a nontrivial elementary embedding $$\hat{j}$$ of $$M$$ into itself with $$\alpha. 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).
• The question is definitely in the context of ZF, not ZFC. In ZFC we already have a problem by taking $M=V_{\kappa+2}$. – Wojowu Jan 2 '19 at 19:09
• it is much shorter to say $\kappa \subseteq M$ than saying $M$ is transitive and $\kappa \in M$. What remain is the class condition? – Zuhair Al-Johar Jan 2 '19 at 19:36
• @ZuhairAl-Johar Yes, but I don't think efficiency is always the highest goal. Focusing explicitly on transitive sets makes the picture easier to think about, in my opinion; we don't have to pay attention to Mostowski collapses. It also plays more nicely with other notions where transitivity/$\ni\kappa$ is more important. Re: the class condition, like I said (in my edit) I don't know. – Noah Schweber Jan 2 '19 at 19:38