Английская Википедия:Huge cardinal

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In mathematics, a cardinal number <math>\kappa</math> is called huge if there exists an elementary embedding <math>j : V \to M</math> from <math>V</math> into a transitive inner model <math>M</math> with critical point <math>\kappa</math> and

<math>{}^{j(\kappa)}M \subset M.</math>

Here, <math>{}^\alpha M</math> is the class of all sequences of length <math>\alpha</math> whose elements are in <math>M</math>.

Huge cardinals were introduced by Шаблон:Harvs.

Variants

In what follows, <math>j^n</math> refers to the <math>n</math>-th iterate of the elementary embedding <math>j</math>, that is, <math>j</math> composed with itself <math>n</math> times, for a finite ordinal <math>n</math>. Also, <math>{}^{<\alpha}M</math> is the class of all sequences of length less than <math>\alpha</math> whose elements are in <math>M</math>. Notice that for the "super" versions, <math>\gamma</math> should be less than <math>j(\kappa)</math>, not <math>{j^n(\kappa)}</math>.

κ is almost n-huge if and only if there is <math>j : V \to M</math> with critical point <math>\kappa</math> and

<math>{}^{<j^n(\kappa)}M \subset M.</math>

κ is super almost n-huge if and only if for every ordinal γ there is <math>j : V \to M</math> with critical point <math>\kappa</math>, <math>\gamma< j(\kappa)</math>, and

<math>{}^{<j^n(\kappa)}M \subset M.</math>

κ is n-huge if and only if there is <math>j : V \to M</math> with critical point <math>\kappa</math> and

<math>{}^{j^n(\kappa)}M \subset M.</math>

κ is super n-huge if and only if for every ordinal <math>\gamma</math> there is <math>j : V \to M</math> with critical point <math>\kappa</math>, <math>\gamma< j(\kappa)</math>, and

<math>{}^{j^n(\kappa)}M \subset M.</math>

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 <math>n</math>-huge for all finite <math>n</math>.

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.

Kanamori, Reinhardt, and Solovay defined seven large cardinal properties between extendibility and hugeness in strength, named <math>\mathbf A_2(\kappa)</math> through <math>\mathbf A_7(\kappa)</math>, and a property <math>\mathbf A_6^\ast(\kappa)</math>. The additional property <math>\mathbf A_1(\kappa)</math> is equivalent to "<math>\kappa</math> is huge", and <math>\mathbf A_3(\kappa)</math> is equivalent to "<math>\kappa</math> is <math>\lambda</math>-supercompact for all <math>\lambda<j(\kappa)</math>". Corazza introduced the property <math>A_{3.5}</math>, lying strictly between <math>A_3</math> and <math>A_4</math>.

Consistency strength

The cardinals are arranged in order of increasing consistency strength as follows:

• almost <math>n</math>-huge
• super almost <math>n</math>-huge
• <math>n</math>-huge
• super <math>n</math>-huge
• almost <math>n+1</math>-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 <math>\omega</math>-huge cardinal <math>\kappa</math> as one such that an elementary embedding <math>j : V \to M</math> from <math>V</math> into a transitive inner model <math>M</math> with critical point <math>\kappa</math> and <math>{}^\lambda M\subseteq M</math>, where <math>\lambda</math> is the supremum of <math>j^n(\kappa)</math> for positive integers <math>n</math>. 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 <math>\omega</math>-huge cardinal <math>\kappa</math> is defined as the critical point of an elementary embedding from some rank <math>V_{\lambda+1}</math> 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

• Шаблон:Citation.
• Шаблон:Citation.
• Шаблон:Citation. A copy of parts I and II of this article with corrections is available at the author's web page.