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".



Upward reflection of rank-into-rank cardinals
The answer is negative. 


The existence of a rank-to-rank cardinal $j:V\_\lambda\to V\_\lambda$ is $\Sigma\_2$ expressible, since it is witnessed inside any sufficiently large $V\_\alpha$. Therefore, if one cuts off at any inaccessible or worldly cardinal above $\lambda$, one still has the rank-to-rank cardinal, but there would be no large cardinals above $\lambda$. 


So the answers to questions 1 and 3 are strongly negative. Basically, if $\kappa$ is rank-to-rank, witnessed by embedding $j:V\_\lambda\to V\_\lambda$ with critical point $\kappa$, then $\lambda$ is the limit of the critical sequence, and this gives you $\omega$ many additional rank-to-rank cardinals. But by cutting off above $\lambda$, one can see that $\lambda$ itself is the limit of any upward reflection phenomenon. 


In a sense, this perspective shows that with rank-to-rank cardinals, it is not necessarily the critical point $\kappa$ that is important, but the cardinal $\lambda$ that is relevant. And this way of thinking destroys the upward-reflection idea, since $\lambda$ is not reflecting upward at all.


Meanwhile, the cardinal $\lambda$ itself must be worldly of very high order, since it is the union of an elementary chain of rank-to-rank cardinals. This can be seen as a positive answer to question 2.


