Kunen showed that Reinhardt cardinals are inconsistent in ZFC. But his proof is a bit technical for a non-set-theorist to follow. Berkeley cardinals are stronger than Reinhardt cardinals. You can refute them in ZFC by observing that every Berkeley cardinal is Reinhardt, and then appealing to Kunen's theorem.

**Question 1:** Is there a more direct refutation of Berkeley cardinals in ZFC, perhaps one which might be more digestible by a non-set-theorist?

Recall that a definably-Berkeley cardinal is a cardinal $\kappa$ such that whenever $M$ is a transitive set with $\kappa \in M$, there are elementary embeddings $M \to M$ with arbitrarily large critical point $\alpha < \kappa$. A Reinhardt cardinal is when this holds for $M = V_\kappa$ (EDIT: In fact the "Reinhardt cardinal is defined to be the critical point of the embedding). So perhaps there is some other $M$ one can cook up which obviously fails the Berkeley property? Maybe $M = \kappa + 1$, for example?

(A Berkeley cardinal is a cardinal $\kappa$ such that whenver $M$ is a transitive set with $\kappa \in M$ and $A \subseteq M$, there are elementary embeddings $j : (M,A) \to (M,A)$ with arbitrarily large critical point below $\kappa$. Although it seems sometimes one restricts the definition to apply only when $M = V_\alpha$? I'm not sure if that's equivalent or weaker.)

**Question 2:** Let $\kappa$ be a cardinal (probably uncountable, regular, limit, measurable,etc.). Can it be the case in ZFC that there exists a nontrivial elementary embedding $\kappa + 1 \to \kappa + 1$?

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