Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 116 / 2 / c / Solution

Letbe the supremum of the Kunen critical sequence, and putThe required choices areIndeed $X$ has rank $\hat{\kappa }$, so $X\in V_{\hat{\kappa }+1}$, while the Kunen lemma gives $X\notin M$ once the domain contains the omega-Jonsson function required by the proof, which is ensured by $\alpha \ge \hat{\kappa }+2$.