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 V_λ of the von Neumann hierarchy.)

• Axiom I3: There is a nontrivial elementary embedding of V_λ into itself.
• Axiom I2: 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.
• Axiom I1: There is a nontrivial elementary embedding of V_λ+1 into itself.
• Axiom I0: There is a nontrivial elementary embedding of L(V_λ+1) into itself with the critical point below λ.

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 V_α 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.

Every I0 cardinal κ (speaking here of the critical point of j) is an I1 cardinal.

Every I1 cardinal κ (sometimes called ω-huge cardinals) is an I2 cardinal and has a stationary set of I2 cardinals below it.

Every I2 cardinal κ is an I3 cardinal and has a stationary set of I3 cardinals below it.

Every I3 cardinal κ has another I3 cardinal above it and is an n-huge cardinal for every n<ω.

Axiom I1 implies that V_λ+1 (equivalently, H(λ⁺)) does not satisfy V=HOD. There is no set S⊂λ definable in V_λ+1 (even from parameters V_λ and ordinals <λ⁺) with S cofinal in λ and |S|<λ, that is, no such S witnesses that λ is singular. And similarly for Axiom I0 and ordinal definability in L(V_λ+1) (even from parameters in V_λ). However globally, and even in V_λ,^[1] V=HOD is relatively consistent with Axiom I1.

References

• Gaifman, Haim (1974), "Elementary embeddings of models of set-theory and certain subtheories", Axiomatic set theory, Proc. Sympos. Pure Math., XIII, Part II, Providence R.I.: Amer. Math. Soc., pp. 33–101, MR 0376347 
• Kanamori, Akihiro (2003), The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nd ed.), Springer, ISBN 3-540-00384-3 
• Laver, Richard (1997), "Implications between strong large cardinal axioms", Ann. Pure Appl. Logic, 90 (1-3): 79–90, MR 1489305, doi:10.1016/S0168-0072(97)00031-6 
• Solovay, Robert M.; Reinhardt, William N.; Kanamori, Akihiro (1978), "Strong axioms of infinity and elementary embeddings", Annals of Mathematical Logic, 13 (1): 73–116, doi:10.1016/0003-4843(78)90031-1