TITLE: What can the extremely large cardinals tell us about small sets?
QUESTION [11 upvotes]: 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 $(\aleph_3,\aleph_2,\aleph_1) \twoheadrightarrow  (\aleph_2,\aleph_1,\aleph_0)$.
I would be remiss not to mention the results of Laver about left-distributive algebras proved using Axiom I3.  Some of the results have not been brought down to ZFC.  This is remarkable and I suppose answers the question as posed, but my motivation is to see if there are forcing constructions aimed at small cardinals that use these very large cardinals.

REPLY [9 votes]: The paper Generic $I_0$ at $\aleph_\omega$ by Vincenzo Dimonte might be of interest to you. It introduces the notion of being generic $I_0$ at $\aleph_\omega$ (Def. 3.1 of the paper), and proves several consequences of it. In particular it is shown that if eneric $I_0$ holds at $\aleph_\omega$, then $\aleph_\omega$ is Jonsson.
Of course it is open if generic $I_0$ at $\aleph_\omega$ can be consistent!.