Huge cardinal
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In mathematics, a cardinal number κ is called huge if and only 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.
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[edit] Variants
In what follows, jn refers to the n-th iterate of the elementary embedding j, that is, j composed with itself n times. Also, <αM is the class of all sequences of length less than α whose elements are in M. Notice that for the "super" versions, λ should be less than j(κ), not jn(κ). And n must be a finite ordinal (ω-huge would be a Reinhardt cardinal and inconsistent with ZFC).
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 λ there is j : V → M with critical point κ, λ<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 λ there is j : V → M with critical point κ, λ<j(κ), and
Notice that 0-huge is the same as measurable cardinal; and 1-huge is the same as huge.
[edit] Consistency strength
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
[edit] See also
[edit] References
- 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, Springer. ISBN 3540003843.
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