Huge cardinal
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In mathematics, a cardinal number κ is called huge iff there exists an elementary embedding j : V → M from V into a transitive inner model M with critical point κ and
- <math>{}^{j(\kappa)}M \subset M.<math>
Here, αM is the class of all sequences of length α whose elements are in M.
κ is almost huge iff there is j : V → M with critical point κ and
- <math>{}^{
Here, <αM is the class of all sequences of length less than α whose elements are in M.
κ is superhuge iff for every ordinal λ there is j : V → M with critical point κ, λ<j(κ), and
- <math>{}^{j(\kappa)}M \subset M.<math>
κ is super almost huge iff for every ordinal λ there is j : V → M with critical point κ, λ<j(κ), and
- <math>{}^{
κ is n-huge iff there is j : V → M with critical point κ and
- <math>{}^{j^n(\kappa)}M \subset M.<math>
Here, jn refers to the n-th iterate of the function j, that is, j composed with itself n times.
Super n-huge and super almost n-huge cardinals are defined analogously. For the "super" versions, λ should be less than j(κ), not j^n(κ). The cardinals are arranged in order of increasing consistency strength as follows:
- almost n-huge
- super almost n-huge
- n-huge
- super n-huge
- almost n+1-huge