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.

Huge cardinals were introduced by Kenneth Kunen in 1972.

Contents 1 Variants 2 Consistency strength 3 See also 4 References

[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, 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, λ should be less than j(κ), not jⁿ(κ).

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. A cardinal satisfying one of the rank into rank axioms is n-huge for all finite n.

The existence of a huge cardinal implies that Vopenka's principle is consistent.

[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

• List of large cardinal properties

[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 3-540-00384-3.