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

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