Rank-into-rank
Encyclopedia
In set theory
, a branch of mathematics
, a rank-into-rank is a large cardinal λ satisfying one of the following four axiom
s (commonly known as rank-into-rank embeddings, given in order of increasing consistency strength):
These are essentially the strongest known large cardinal axioms not known to be inconsistent in ZFC; the axiom for Reinhardt cardinal
s 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 Vα 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 cardinal
s 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 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.
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
, a branch of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a rank-into-rank is a large cardinal λ satisfying one of the following four axiom
Axiom
In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...
s (commonly known as rank-into-rank embeddings, given in order of increasing consistency strength):
- Axiom I3: There is a nontrivial elementary embedding of Vλ into itself.
- Axiom 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 pointCritical point (set theory)In set theory, the critical point of an elementary embedding of a transitive class into another transitive class is the smallest ordinal which is not mapped to itself....
. - Axiom I1: There is a nontrivial elementary embedding of Vλ+1 into itself.
- Axiom I0: There is a nontrivial elementary embedding of L(Vλ+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 cardinal
Reinhardt cardinal
In set theory, a branch of mathematics, a Reinhardt cardinal is a large cardinal κ, suggested by , that is the critical point of a non-trivial elementary embedding j of V into itself....
s 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
Critical point (set theory)
In set theory, the critical point of an elementary embedding of a transitive class into another transitive class is the smallest ordinal which is not mapped to itself....
, 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 Vα into itself then α is either a limit ordinal of cofinalityCofinality
In mathematics, especially in order theory, the cofinality cf of a partially ordered set A is the least of the cardinalities of the cofinal subsets of A....
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
Kunen's inconsistency theorem
In set theory, a branch of mathematics, Kunen's inconsistency theorem, proved by , shows that several plausible large cardinal axioms are inconsistent with the axiom of choice.Some consequences of Kunen's theorem are:...
that Reinhardt cardinal
Reinhardt cardinal
In set theory, a branch of mathematics, a Reinhardt cardinal is a large cardinal κ, suggested by , that is the critical point of a non-trivial elementary embedding j of V into itself....
s 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 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.