Huge cardinal

In mathematics, a cardinal number κ is called huge if there exists an elementary embedding j : V → M from V into a transitive inner model M with critical point κ and

Here, ^αM is the class of all sequences of length α whose elements are in M.

Huge cardinals were introduced by Kenneth Kunen (1978).

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 ${\displaystyle {j^{n}(\kappa )}}$.

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

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 n-huge for all finite n.

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