In mathematics, a cardinal number
is called huge if there exists an elementary embedding
into a transitive inner model
with critical point
is the class of all sequences of length
Huge cardinals were introduced by Kenneth Kunen (1978).
-th iterate of the elementary embedding
times, for a finite ordinal
is the class of all sequences of length less than
Notice that for the "super" versions,
κ is almost n-huge if and only if there is
with critical point
and κ is super almost n-huge if and only if for every ordinal γ there is
with critical point
γ < j ( κ )
, and κ is n-huge if and only if there is
and κ is super n-huge if and only if for every ordinal
, and Notice that 0-huge is the same as measurable cardinal; and 1-huge is the same as huge.
A cardinal satisfying one of the rank into rank axioms is
The existence of an almost huge cardinal implies that Vopěnka's principle is consistent; more precisely any almost huge cardinal is also a Vopěnka cardinal.
Kanamori, Reinhardt, and Solovay defined seven large cardinal properties between extendibility and hugeness in strength, named
λ < j ( κ )
Corazza introduced the property
, lying strictly between
[2] The cardinals are arranged in order of increasing consistency strength as follows: The consistency of a huge cardinal implies the consistency of a supercompact cardinal, nevertheless, the least huge cardinal is smaller than the least supercompact cardinal (assuming both exist).
as one such that an elementary embedding
into a transitive inner model
However Kunen's inconsistency theorem shows that such cardinals are inconsistent in ZFC, though it is still open whether they are consistent in ZF.
is defined as the critical point of an elementary embedding from some rank
This is closely related to the rank-into-rank axiom I1.