[FOM] role of large cardinals

[meskew at math.uci.edu]

> I0 does not directly contradict Choice, and
> it is stronger than ZF+Reinhardt simply because it says something about
> the structure of this self-embeddable class.

Sorry, I was wrong here.  I0 does not require that one has access to j as
a predicate from within L(V_{\lambda+1}).  Indeed, ZF+Reinhardt implies
Con(ZFC+I0).

However, I think Reinhardt cardinals would still be considered large
cardinals by set theorists.  What else would you call them?  It would be
interesting to see a hypothesis of an entirely different flavor that
implies the consistency of the choiceless large cardinals.