In set theory, a branch of mathematics, a rank-into-rank embedding is a large cardinal property defined by one of the following four axioms given in order of increasing consistency strength.
(A set of rank
is one of the elements of the set
of the von Neumann hierarchy.)
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.
is the elementary embedding mentioned in one of these axioms and
is its critical point, then
λ
More generally, if the axiom of choice holds, it is provable that if there is a nontrivial elementary embedding of
is either a limit ordinal of cofinality
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 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.
(speaking here of the critical point of
(sometimes called ω-huge cardinals) is an I2 cardinal and has a stationary set of I2 cardinals below it.
is an I3 cardinal and has a stationary set of I3 cardinals below it.
Axiom I1 implies that
λ + 1
λ
⊂ λ
And similarly for Axiom I0 and ordinal definability in
,[1] V=HOD is relatively consistent with Axiom I1.
Notice that I0 is sometimes strengthened further by adding an "Icarus set", so that it would be The Icarus set should be in
but chosen to avoid creating an inconsistency.
So for example, it cannot encode a well-ordering of
See section 10 of Dimonte for more details.
Woodin defined a sequence of sets
for use as Icarus sets.