ⓘ Rankintorank
In set theory, a branch of mathematics, a rankintorank embedding is a large cardinal property defined by one of the following four axioms given in order of increasing consistency strength.
 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 I3: There is a nontrivial elementary embedding of V λ into itself.
 Axiom I0: There is a nontrivial elementary embedding of LV λ+1 into itself with 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 j n κ {\displaystyle j^{n}\kappa} 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 I0, I1, I2, and I3 were at first suspected to be inconsistent in ZFC as it was thought possible that Kunens 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 ωhuge 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
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Large cardinal 
Axiom of determinacy 

Berkeley cardinal 
Core model 
Critical point (set theory) 
Extender (set theory) 

Extendible cardinal 
Grothendieck universe 
Huge cardinal 
Indescribable cardinal 
Ineffable cardinal 
Iterable cardinal 
Kunens inconsistency theorem 
Mahlo cardinal 
Measurable cardinal 
Reinhardt cardinal 
Remarkable cardinal 
Shelah cardinal 
Shrewd cardinal 
Strong cardinal 
Strongly compact cardinal 
Quasicompact cardinal 
Unfoldable cardinal 
Weakly compact cardinal 
Wholeness axiom 
Woodin cardinal 
Zero sharp 

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