ω-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

:$^M \subset M.\!$

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

Huge cardinals were introduced by .

Variants

In what follows, j^''n'' 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 $$.

κ is almost n-huge if and only if there is ''j'' : ''V'' → ''M'' with critical point κ and

:$^M \subset M.\!$

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

:$^M \subset M.\!$

κ is n-huge if and only if there is ''j'' : ''V'' → ''M'' with critical point κ and

:$^M \subset M.\!$

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

:$^M \subset M.\!$

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 an almost huge cardinal implies that Vopěnka's principle is consistent; more precisely any almost huge cardinal is also a Vopěnka cardinal.

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
The consistency of a huge cardinal implies the consistency of a supercompact cardinal, nevertheless, the least huge cardinal is smaller than the least supercompact cardinal (assuming both exist).

ω-huge cardinals

One can try defining an ω-huge cardinal κ as one such that an elementary embedding j : V → M from V into a transitive inner model M with critical point κ and ^λ''M''⊆''M'', where λ is the supremum of ''j''^''n''(κ) for positive integers ''n''. However Kunen's inconsistency theorem shows that such cardinals are inconsistent in ZFC, though it is still open whether they are consistent in ZF. Instead an ω-huge cardinal κ is defined as the critical point of an elementary embedding from some rank ''V''_λ+1 to itself. This is closely related to the rank-into-rank axiom I₁.

See also

* List of large cardinal properties
*The Dehornoy order on a braid group was motivated by properties of huge cardinals.

References

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*{{Citation, last=Maddy, first=Penelope, authorlink=Penelope Maddy, journal=The Journal of Symbolic Logic, title=Believing the Axioms. II, year=1988, volume=53, issue=3, pages=736-764 (esp. 754-756), doi=10.2307/2274569, jstor=2274569, s2cid=16544090 , url=https://semanticscholar.org/paper/8d3d986c97fa971246dffd101c411d4e071c4155. A copy of parts I and II of this article with corrections is available at th
author's web page

Large cardinals