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 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 j n ( κ ) {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.