Question about rank into rank cardinals I do not understand why λ is smaller than $\kappa_0$ if λ is the supremum of a growing sequence that starts at $\kappa_0$ (see definition below).
From cantor's attic (http://cantorsattic.info/Rank_1_into_rank_1 , second pargraph):
"There are really two cardinals relevant to such embeddings: The large cardinal is the critical point of j, often denoted cr(j) or sometimes $\kappa_0$, and the other (not quite so large) cardinal is λ. The cardinal λ is determined by defining the critical sequence of j. Set $\kappa_0$=cr(j) and $\kappa_{n+1}$=j($\kappa_n$). 
Then λ=sup$\langle \kappa_n:n<ω\rangle$ and is the first fixed point of j that occurs above $\kappa_0$"

The text means to say that $\kappa_0$ is a large cardinal. It's the critical point of an elementary embedding, and therefore at least measurable. On the other hand, $\lambda$ has cofinality $\omega$ making it not even inaccessible -- something required from most large cardinals.
But recall that we have $\kappa_0<j(\kappa_0)=\kappa_1<\ldots<j(\kappa_n)=\kappa_{n+1}<\ldots<\lambda$. So in terms of cardinality per se, $\lambda$ can be much much much larger than $\kappa_0$.
