Rank-into-rank
In
set theory, the following four
axioms, commonly known as
rank-into-rank embeddings, are among the most powerful
large cardinal axioms known. Stated in the order of increasing strength, they are as follows:
- I3: There is a nontrivial elementary embedding of Vλ into itself.
- I2: There is a nontrivial elementary embedding of V into a transitive class M that includes Vλ where λ is the first fixed point above the critical point.
- I1: There is a nontrivial elementary embedding of Vλ+1 into itself.
- I0: There is a nontrivial elementary embedding of L(Vλ+1 ) into itself with the critical point below λ.
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.