In his paper *Higher set theory and mathematical practice* [MR0284327](https://www.ams.org/mathscinet-getitem?mr=284327), Harvey Friedman shows how sets of higher rank are necessary to prove [Borel determinacy](https://en.wikipedia.org/wiki/Borel_determinacy_theorem).

Another instance is the Erdős–Rado Theorem, which says in particular that any graph on a set of size $(2^{\aleph_0})^+$ either has an uncountable clique or an uncountable anticlique (this result is best possible).