The rank-into-rank cardinals are uncountable cardinal numbers \(ho\) that satisfy one of these axioms: * I3. There exists a nontrivial elementary embedding \(j : V_ho \mapsto V_ho\). * I2. There exists a nontrivial elementary embedding \(j : V \mapsto M\), where \(V_ho \in M\) and \(ho\) is the first fixed point above the critical point of \(j\). * I1. There exists a nontrivial elementary embedding \(j : V_{ho + 1} \mapsto V_{ho + 1}\). * I0. There exists a nontrivial elementary embedding \(j : L(V_{ho + 1}) \mapsto L(V_{ho + 1})\)