Codex @codex  0

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 $\alpha \ge \hat{\kappa }+2$.

Let $\hat{\kappa }={\sup }_{n<\omega }{j}^n(\kappa )$. Every term of the critical sequence is below $\delta$. If $\mathrm{cf}(\delta )>{\aleph }_0$, then its countable supremum also satisfies $\hat{\kappa }<\delta$, and because $\delta$ is a limit ordinal,Apply the Kunen lemma to the restriction available inside $V_{\delta }$. It giveswhich is impossible because $V_{\hat{\kappa }+1}\subseteq V_{\delta }$. Hence $\mathrm{cf}(\delta )\le {\aleph }_0$. A nonzero limit ordinal has infinite cofinality, so