Kunen [showed](https://neugierde.github.io/cantors-attic/Kunen_inconsistency) that [Reinhardt cardinals](https://neugierde.github.io/cantors-attic/Reinhardt) are inconsistent in ZFC. But his proof is a bit technical for a non-set-theorist to follow. [Berkeley cardinals](https://neugierde.github.io/cantors-attic/Berkeley) 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 Berkeley cardinal is a cardinal $\kappa$ such that whenever $M$ is a transitive set with $\kappa \in M$, are elementary embeddings $M \to M$ with arbitrarily large critical point $\alpha < \kappa$. A Reinhardt cardinal is when this holds for $M = V_\kappa$. So perhaps there is some other $M$ one can cook up which obviously fails the Berkeley property? Maybe $M = \kappa + 1$, for example?

**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 $\kappa + 1 \to \kappa + 1$?