In what follows, refers to the -th iterate of the elementary embedding , that is, composed with itself times, for a finite ordinal . Also, is the class of all sequences of length less than whose elements are in . Notice that for the "super" versions, should be less than , not .
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 , , and
κ is n-huge if and only if there is with critical point and
κ is super n-huge if and only if for every ordinal there is with critical point , , 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 -huge for all finite .
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 through , and a property .[1] The additional property is equivalent to " is huge", and is equivalent to " is -supercompact for all ". Corazza introduced the property , lying strictly between and .[2]
Consistency strength
The cardinals are arranged in order of increasing consistency strength as follows:
almost -huge
super almost -huge
-huge
super -huge
almost -huge
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).
One can try defining an -huge cardinal as one such that an elementary embedding from into a transitive inner model with critical point and , where is the supremum of for positive integers . However Kunen's inconsistency theorem shows that such cardinals are inconsistent in ZFC, though it is still open whether they are consistent in ZF. Instead an -huge cardinal is defined as the critical point of an elementary embedding from some rank to itself. This is closely related to the rank-into-rank axiom I1.