(Why is Kunen inconsistency at the top of Cantor's upper attic?) I have seen statements about the Kunen Inconsistency being thrown around like that it is the 'upper-bound' or 'limit' for how strong of a large cardinal you can create within ZFC (Kunen inconsistency cardinals are inconsistent with ZFC). This implies that these cardinals (Reinhardt, ω-huge cardinals, etc.) are too 'strong' for choice and that there mere implication 'break choice.' This may apply to Reinhardt and Berkeley cardinals, but what about ω-huge cardinals?
As seen from Definition of an $\omega$-huge cardinal , ω-huge cardinals are equivalent to the large cardinal axiom I1 in strength (they are also both constructed via a elementary embedding from some rank Vλ+1 to itself), below the likes of I0 (which is consistent with ZFC). Here is my question: Why are ω-huge cardinals below the choiceless cardinals (Reinhardt, etc.) when it is apart of the Kunen Inconsistency and are supposed to 'too strong' for choice?
My current theory (with my limited knowledge) rather is that the 'Kunen Inconsistency isn't a limit, but rather a inconsistency occurs when the schema for these large cardinals are found within ZFC (Like they are 'incompatible' with the axiom of choice). Is the Kunen inconsistency really a limit or not?