In his paper Higher set theory and mathematical practice MR0284327, Harvey Friedman shows how sets of higher rank are necessary to prove Borel determinacy.

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).

In his paper Higher set theory and mathematical practice MR0284327, Harvey Friedman shows how sets of higher rank are necessary to prove Borel determinacy.

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).

In his paper Higher set theory and mathematical practice MR0284327, Harvey Friedman shows how sets of higher rank are necessary to prove Borel determinacy.

In his paper Higher set theory and mathematical practice MR0284327, Harvey Friedman shows how sets of higher rank are necessary to prove Borel determinacy.

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).