Rank-into-rank cardinals have the rather intriguing property that they reflect upwards. I would be interested to know how far the upward reflection goes:
1) Does "There exists a rank-into-rank cardinal (of type I3, say)" imply that "There exists an unbounded class of rank-into-rank cardinals (in V) ?"
2) If 1) is false, then can we say something interesting about the supremum of this upward reflecting sequence ? For instance, maybe the supremum has some large cardinal properties of interest ?
3) Finally, existence of a rank-into-rank implies the existence of pretty much all of the other large cardinals. But can we strengthen this along the lines of "If there exists a rank-into-rank cardinal, then there exists an unbounded class of cardinals with property P (in V)" for interesting some property P? By interesting property P, I mean something like "inaccessible" or "Mahlo" or hopefully even stronger like "measurable".