If \(\kappa\) is measurable and \(j:V\to M\) is given by the ultrapower construction, then \(V_\kappa \subseteq M\), and every sequence of length less than or equal to \(\kappa\) of elements of \(M\) belongs to \(M\). Thus, \(M\) is quite similar to \(V\), but it cannot be \(V\) itself. Indeed, a famous theorem of Kenneth Kunen shows that there cannot be any elementary embedding \(j:V\to V\), other than the trivial one, i.e., the identity. All known proofs of this result use the Axiom of Choice, and it is an outstanding important question if the axiom is necessary. Kunen’s Theorem opens the door to formulating large cardinal notions stronger than measurability by requiring that \(M\) is closer to \(V\).