Are there any applications of the largest large cardinals to consistency results concerning, say, cardinals below $\aleph_{\aleph_\omega}$? Or perhaps to prove results in descriptive set theory? I am thinking of ZFC + Axiom I0, ZF + Reinhardt, ZF + Berkeley. See here for definitions. The largest cardinal I know of to be used for a consistency result is a 2-huge cardinal to prove the consistency of $(\aleph_3,\aleph_2,\aleph_1) \twoheadrightarrow (\aleph_2,\aleph_1,\aleph_0)$.
What can the extremely large cardinals tell us about small sets?
Monroe Eskew
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