Rank-into-rank Guide, Meaning , Facts, Information and Description

In set theory, a rank-into-rank is a large cardinal λ satisfying one of the following four axioms (commonly known as rank-into-rank embeddings, given in order of increasing consistency strength):

1. There is a nontrivial elementary embedding of V_λ into itself.
2. There is a nontrivial elementary embedding of V into a transitive class M that includes V_λ where λ is the first fixed point above the critical point.
3. There is a nontrivial elementary embedding of V_λ+1 into itself.
4. There is a nontrivial elementary embedding of L(V_λ+1 ) into itself with the critical point below λ.

The axioms are called I3, I2, I1, and I0 respectively. Assuming the axiom of choice, it is provable that if there is a nontrivial elementary embedding of V_λ into itself then λ is a limit ordinal of cofinality ω or the successor of such an ordinal.

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