Huge cardinal

In mathematics, a cardinal number ${\displaystyle \kappa }$ is called huge if there exists an elementary embedding ${\displaystyle j:V\to M}$ from ${\displaystyle V}$ into a transitive inner model ${\displaystyle M}$ with critical point ${\displaystyle \kappa }$ and

${\displaystyle {}^{j(\kappa )}M\subset M.}$

Here, ${\displaystyle {}^{\alpha }M}$ is the class of all sequences of length ${\displaystyle \alpha }$ whose elements are in ${\displaystyle M}$.

Huge cardinals were introduced by Kenneth Kunen (1978).

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

κ is almost n-huge if and only if there is ${\displaystyle j:V\to M}$ with critical point ${\displaystyle \kappa }$ and

${\displaystyle {}^{<j^{n}(\kappa )}M\subset M.}$

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

${\displaystyle {}^{<j^{n}(\kappa )}M\subset M.}$

κ is n-huge if and only if there is ${\displaystyle j:V\to M}$ with critical point ${\displaystyle \kappa }$ and

${\displaystyle {}^{j^{n}(\kappa )}M\subset M.}$

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

${\displaystyle {}^{j^{n}(\kappa )}M\subset M.}$

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 ${\displaystyle n}$-huge for all finite ${\displaystyle n}$.

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.

Kanamori, Reinhardt, and Solovay defined seven large cardinal properties between extendibility and hugeness in strength, named ${\displaystyle \mathbf {A} _{2}(\kappa )}$ through ${\displaystyle \mathbf {A} _{7}(\kappa )}$, and a property ${\displaystyle \mathbf {A} _{6}^{\ast }(\kappa )}$.^[1] The additional property ${\displaystyle \mathbf {A} _{1}(\kappa )}$ is equivalent to "${\displaystyle \kappa }$ is huge", and ${\displaystyle \mathbf {A} _{3}(\kappa )}$ is equivalent to "${\displaystyle \kappa }$ is ${\displaystyle \lambda }$-supercompact for all ${\displaystyle \lambda <j(\kappa )}$". Corazza introduced the property ${\displaystyle A_{3.5}}$, lying strictly between ${\displaystyle A_{3}}$ and ${\displaystyle A_{4}}$.^[2]

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

• almost ${\displaystyle n}$-huge
• super almost ${\displaystyle n}$-huge
• ${\displaystyle n}$-huge
• super ${\displaystyle n}$-huge
• almost ${\displaystyle n+1}$-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).

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One can try defining an ${\displaystyle \omega }$-huge cardinal ${\displaystyle \kappa }$ as one such that an elementary embedding ${\displaystyle j:V\to M}$ from ${\displaystyle V}$ into a transitive inner model ${\displaystyle M}$ with critical point ${\displaystyle \kappa }$ and ${\displaystyle {}^{\lambda }M\subseteq M}$, where ${\displaystyle \lambda }$ is the supremum of ${\displaystyle j^{n}(\kappa )}$ for positive integers ${\displaystyle n}$. 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 ${\displaystyle \omega }$-huge cardinal ${\displaystyle \kappa }$ is defined as the critical point of an elementary embedding from some rank ${\displaystyle V_{\lambda +1}}$ to itself. This is closely related to the rank-into-rank axiom I₁.

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