Rank-into-rank cardinal

The rank-into-rank cardinals are one of the strongest known classes of uncountable cardinal. They are defined as precisely the uncountable cardinals \(\rho\) that satisfy one of these axioms:


 * I3. There exists an elementary embedding \(j : V_\rho \maptso V_\rho\).
 * I2. There exists an elementary embedding \(j : V \mapsto M\), where \(V_\rho \in M\) and \(\rho\) is the first fixed point above the critical point of \(j\).
 * I1. There exists an elementary embedding \(j : V_{\rho + 1} \mapsto V_{\rho + 1}\).

The axiom asserting the existence of the rank-into-rank cardinals is very strong. It is not provable in, and ZFC + "there exists a rank-into-rank cardinal" is believed to be consistent (although there is reason to doubt this).