Recently I've been thinking about elementary embeddings, partition cardinals, etc. as part of my never-ending quest for understanding of consistency strength :p
I came up with this idea, called I* cardinals. I know their existence implies the existence of I3 cardinals, but is the existence of an I* cardinal equivalent to any existing LCAs and, if so, which ones? I have a hunch from their definition that they may be related to partition cardinals, but I don't know of any partition conditions (consistent with AC) at this level. If I* cardinals aren't equivalent to any existing LCAs, are they at least consistent (relative to other, stronger LCAs)? Or are they plain inconsistent?
For an ordinal $\alpha$, $\alpha^*$ is the next $\Sigma_\omega$-sound cardinal after $\alpha$.
A set $X$ is a $Y$-elementary system if, for all $\mathcal{X}, \mathcal{Y} \subseteq X$ with $|\mathcal{X}| = |\mathcal{Y}| < \aleph_0$, there is a nontrivial elementary embedding $j_{\mathcal{X}, \mathcal{Y}}: Y \to Y$ so that $j_{\mathcal{X}, \mathcal{Y}}''\mathcal{X} = \mathcal{Y}$.
A cardinal $\kappa$ is an I* cardinal if there is a set $X$ so that $|X| = \kappa$ and $X \cup \{\kappa\}$ is a $V_{\kappa^*}$-elementary system.
