Kunen's inconsistency theorem

Remarks

Let $j$ be a nontrivial embedding with critical point $\kappa_0$, let $\kappa_{n+1} = j(\kappa_n)$, and let $\lambda = \sup_n \kappa_n$, so that $j(\lambda) = \lambda$. The proof uses the axiom of choice to find a function on the countable subsets of $\lambda$ that $j$ cannot respect. One such tool is an $\omega$-Jónsson function, from a theorem of Erdős and Hajnal. Part 2 follows from part 1 applied to $\delta = \lambda$, after restricting $j$ to $V_{\lambda + 2}$. The theorem sets an upper limit to large cardinal axioms built from embeddings: the axioms just below it ask for elementary embeddings of $V_\lambda$ or $V_{\lambda + 1}$ into itself. It is an open problem whether part 2 holds without the axiom of choice.

Going deeper

For a student

The embeddings $j \colon V \to M$ of measurable and supercompact cardinals become stronger as $M$ gets closer to $V$. Kunen's theorem says that $M$ cannot be all of $V$, and more precisely that $M$ cannot contain the set $j[\lambda]$ for the $\lambda$ above. The strongest large cardinal axioms in use, such as elementary embeddings $j \colon V_{\lambda + 1} \to V_{\lambda + 1}$, sit just below the limit given by part 1.