Here, αM is the class of all sequences of length α whose elements are in M.
In what follows, jn refers to the n-th iterate of the elementary embedding j, that is, j composed with itself n times, for a finite ordinal n. Also, <αM is the class of all sequences of length less than α whose elements are in M. Notice that for the "super" versions, γ should be less than j(κ), not .
κ is almost n-huge if and only if there is j : V → M with critical point κ and
κ is super almost n-huge if and only if for every ordinal γ there is j : V → M with critical point κ, γ<j(κ), and
κ is n-huge if and only if there is j : V → M with critical point κ and
κ is super n-huge if and only if for every ordinal γ there is j : V → M with critical point κ, γ<j(κ), and
The cardinals are arranged in order of increasing consistency strength as follows:
- almost n-huge
- super almost n-huge
- super n-huge
- almost n+1-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 ω-huge cardinal κ as one such that an elementary embedding j : V → M from V into a transitive inner model M with critical point κ and λM⊆M, where λ is the supremum of jn(κ) for positive integers n. However Kunen's inconsistency theorem shows that ω-huge cardinals are inconsistent in ZFC, though it is still open whether they are consistent in ZF.
- List of large cardinal properties
- The Dehornoy order on a braid group was motivated by properties of huge cardinals.
- Kunen, Kenneth (1978), "Saturated ideals", The Journal of Symbolic Logic 43 (1): 65–76, doi:10.2307/2271949, ISSN 0022-4812, MR 495118
- Penelope Maddy,"Believing the Axioms,II"(i.e. part 2 of 2),"Journal of Symbolic Logic",vol.53,no.3,Sept.1988,pages 736 to 764 (esp.754-756).
- Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nd ed ed.). Springer. ISBN 3-540-00384-3.