Rank-into-rank
In set theory, a branch of mathematics, a rank-into-rank is a large cardinal λ satisfying one of the following four axioms given in order of increasing consistency strength. (They are sometimes known as rank-into-rank embeddings, where a rank is one of the sets Vλ of the von Neumann hierarchy.)
These are essentially the strongest known large cardinal axioms not known to be inconsistent in ZFC; the axiom for Reinhardt cardinals is stronger, but is not consistent with the axiom of choice.
If j is the elementary embedding mentioned in one of these axioms and κ is its critical point, then λ is the limit of as n goes to ω. More generally, if the axiom of choice holds, it is provable that if there is a nontrivial elementary embedding of Vα into itself then α is either a limit ordinal of cofinality ω or the successor of such an ordinal.
The axioms I1, I2, and I3 were at first suspected to be inconsistent (in ZFC) as it was thought possible that Kunen's inconsistency theorem that Reinhardt cardinals are inconsistent with the axiom of choice could be extended to them, but this has not yet happened and they are now usually believed to be consistent.
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