**Huge** cardinals (and their variants) were introduced by Kenneth Kunen
in 1972 as a very large cardinal axiom. Kenneth Kunen first used them to
prove that the consistency of the existence of a huge cardinal implies
the consistency of $\text{ZFC}$+"there is a $\omega_2$-saturated
$\sigma$-[ideal](Filter.md "Filter")
on $\omega_1
quot;. It is now known that only a
[Woodin](Woodin.md "Woodin")
cardinal is needed for this result. However, the consistency of the
existence of an $\omega_2$-complete $\omega_3$-saturated
$\sigma$-ideal on $\omega_2$, as far as the set theory world is
concerned, still requires an almost huge cardinal.
{% cite Kanamori2009 %}
## Definitions
Their formulation is similar to that of the formulation of
[superstrong](Superstrong.md "Superstrong")
cardinals. A huge cardinal is to a
[supercompact](Supercompact.md "Supercompact")
cardinal as a superstrong cardinal is to a
[strong](Strong.md "Strong")
cardinal, more precisely. The definition is part of a generalized
phenomenon known as the "double helix", in which for some large cardinal
properties n-$P_0$ and n-$P_1$, n-$P_0$ has less consistency strength
than n-$P_1$, which has less consistency strength than (n+1)-$P_0$,
and so on. This phenomenon is seen only around the [](N-fold_variants.md)
as of modern set theoretic concerns.
{% cite Kentaro2007 %}
Although they are very large, there is a first-order definition which is
equivalent to n-hugeness, so the $\theta$-th n-huge cardinal is
first-order definable whenever $\theta$ is first-order definable. This
definition can be seen as a (very strong) strengthening of the
first-order definition of
[measurability](Measurable.md "Measurable").
### Elementary embedding definitions
The elementary embedding definitions are somewhat standard. Let
$j:V\rightarrow M$ be a nontrivial [](Elementary%20embedding.md)
of $V$ into a
[transitive](Transitive.md "Transitive")
class $M$ with critical point $\kappa$. Then:
- $\kappa$ is **almost n-huge with target $\lambda$** iff
$\lambda=j^n(\kappa)$ and $M$ is closed under all of its sequences
of length less than $\lambda$ (that is, $M^{<\lambda}\subseteq
M$).
- $\kappa$ is **n-huge with target $\lambda$** iff
$\lambda=j^n(\kappa)$ and $M$ is closed under all of its sequences
of length $\lambda$ ($M^\lambda\subseteq M$).
- $\kappa$ is **almost n-huge** iff it is almost n-huge with target
$\lambda$ for some $\lambda$.
- $\kappa$ is **n-huge** iff it is n-huge with target $\lambda$ for
some $\lambda$.
- $\kappa$ is **super almost n-huge** iff for every $\gamma$, there
is some $\lambda>\gamma$ for which $\kappa$ is almost n-huge
with target $\lambda$ (that is, the target can be made arbitrarily
large).
- $\kappa$ is **super n-huge** iff for every $\gamma$, there is some
$\lambda>\gamma$ for which $\kappa$ is n-huge with target
$\lambda$.
- $\kappa$ is **almost huge**, **huge**, **super almost huge**, and
**superhuge** iff it is **almost 1-huge**, **1-huge**, etc.
respectively.
### Ultrahuge cardinals
A cardinal $\kappa$ is **$\lambda$-ultrahuge** for
$\lambda>\kappa$ if there exists a nontrivial elementary embedding
$j:V\to M$ for some transitive class $M$ such that
$j(\kappa)>\lambda$, $M^{j(\kappa)}\subseteq M$ and
$V_{j(\lambda)}\subseteq M$. A cardinal is **ultrahuge** if it is
$\lambda$-ultrahuge for all $\lambda\geq\kappa$.
<a href="http://logicatorino.altervista.org/slides/150619tsaprounis.pdf" class="external autonumber">[1]</a>
Notice how similar this definition is to the alternative
characterization of
[extendible](Extendible.md "Extendible")
cardinals. Furthermore, this definition can be extended in the obvious
way to define $\lambda$-ultra n-hugeness and ultra n-hugeness, as well
as the "*almost*" variants.
### Hyperhuge cardinals
A cardinal $\kappa$ is **$\lambda$-hyperhuge** for
$\lambda>\kappa$ if there exists a nontrivial elementary embedding
$j:V\to M$ for some inner model $M$ such that $\mathrm{crit}(j) =
\kappa$, $j(\kappa)>\lambda$ and $^{j(\lambda)}M\subseteq M$. A
cardinal is **hyperhuge** if it is $\lambda$-hyperhuge for all
$\lambda>\kappa$. {% cite Usuba2017 Boney2017 %}
### Huge* cardinals
A cardinal $κ$ is **$n$-huge*** if for some $α > κ$, $\kappa$ is
the critical point of an elementary embedding $j : V_α → V_β$ such
that $j^n (κ) <
α$.{% cite Gitmana %}
Hugeness* variant is formulated in a way allowing for a virtual variant
consistent with $V=L$: A cardinal $κ$ is **virtually $n$-huge*** if for
some $α > κ$, in a set-forcing extension, $\kappa$ is the critical
point of an elementary embedding $j : V_α → V_β$ such that $j^n(κ)
< α$.{% cite Gitmana %}
### Ultrafilter definition
The first-order definition of n-huge is somewhat similar to
[measurability](Measurable.md "Measurable").
Specifically, $\kappa$ is measurable iff there is a nonprincipal
$\kappa$-complete
[ultrafilter](Filter.md "Filter"),
$U$, over $\kappa$. A cardinal $\kappa$ is n-huge with target
$\lambda$ iff there is a normal $\kappa$-complete ultrafilter, $U$,
over $\mathcal{P}(\lambda)$, and cardinals
$\kappa=\lambda_0<\lambda_1<\lambda_2...<\lambda_{n-1}<\lambda_n=\lambda$
such that:
$\forall
i<n(\{x\subseteq\lambda:\text{order-type}(x\cap\lambda_{i+1})=\lambda_i\}\in
U)$
Where $\text{order-type}(X)$ is the
[order-type](Order-isomorphism.md "Order-isomorphism")
of the poset $(X,\in)$. {% cite Kanamori2009 %}
$\kappa$ is then super n-huge if for all ordinals $\theta$ there is a
$\lambda>\theta$ such that $\kappa$ is n-huge with target
$\lambda$, i.e. $\lambda_n$ can be made arbitrarily large. If
$j:V\to M$ is such that $M^{j^n(\kappa)}\subseteq M$ (i.e. $j$
witnesses n-hugeness) then there is a ultrafilter $U$ as above such
that, for all $k\leq n$, $\lambda_k = j^k(\kappa)$, i.e. it is not
only $\lambda=\lambda_n$ that is an iterate of $\kappa$ by $j$; all
members of the $\lambda_k$ sequence are.
As an example, $\kappa$ is 1-huge with target $\lambda$ iff there is a
normal $\kappa$-complete ultrafilter, $U$, over
$\mathcal{P}(\lambda)$ such that
$\{x\subseteq\lambda:\text{order-type}(x)=\kappa\}\in U$. The
reason why this would be so surprising is that every set
$x\subseteq\lambda$ with every set of order-type $\kappa$ would be in
the ultrafilter; that is, every set containing
$\{x\subseteq\lambda:\text{order-type}(x)=\kappa\}$ as a subset is
considered a "large set."
As for hyperhugeness, the following are
equivalent:{% cite Boney2017 %}
- $κ$ is $λ$-hyperhuge;
- $μ > λ$ and a normal, fine, κ-complete ultrafilter exists on
$\[μ\]^λ_{∗κ} := \{s ⊂ μ : \|s\| = λ, \|s ∩ κ\| ∈ κ,
\mathrm{otp}(s ∩ λ) < κ\}$;
- $\mathbb{L}_{κ,κ}$ is $\[μ\]^λ_{∗κ}$-$κ$-compact for type
omission.
### Coherent sequence characterization of almost hugeness
### $C^{(n)}$-$m$-huge cardinals
(this section from {% cite Bagaria2012 %})
$κ$ is
**<a href="Correct" class="mw-redirect" title="Correct">$C^{(n)}$-$m$-huge</a>**
iff it is $m$-huge and $j(κ) ∈ C^{(n)}$ ($C^{(n)}$-huge if it is huge
and $j(κ) ∈ C^{(n)}$).
Equivalent definition in terms of normal measures: κ is
$C^{(n)}$-$m$-huge iff it is uncountable and there is a $κ$-complete
normal
<a href="Ultrafilter" class="mw-redirect" title="Ultrafilter">ultrafilter</a>
$U$ over some $P(λ)$ and cardinals $κ = λ_0 < λ_1 < . . . <
λ_m = λ$, with $λ_1 ∈ C (n)$ and such that for each $i < m$, $\{x
∈ P(λ) : ot(x ∩ λ i+1 ) = λ i \} ∈ U$.
It follows that “$κ$ is $C^{(n)}$-$m$-huge” is $Σ_{n+1}$ expressible.
Every huge cardinal is $C^{(1)}$-huge.
The first $C^{(n)}$-$m$-huge cardinal is not $C^{(n+1)}$-$m$-huge, for
all $m$ and $n$ greater or equal than $1$. For suppose $κ$ is the least
$C^{(n)}$-$m$-huge cardinal and $j : V → M$ witnesses that $κ$ is
$C^{(n+1)}$-$m$-huge. Then since “x is $C^{(n)}$-$m$-huge” is $Σ_{n+1}$
expressible, we have $V_{j(κ)} \models$ “$κ$ is $C^{(n)}$-$m$-huge”.
Hence, since $(V_{j(κ)})^M = V_{j(κ)}$, $M \models$ “$∃_{δ <
j(κ)}(V_{j(κ)} \models$ “δ is huge”$)$”. By elementarity, there is a
$C^{(n)}$-$m$-huge cardinal less than $κ$ in $V$ – contradiction.
Assuming [](Rank_into_rank.md),
if $δ$ is a limit cardinal (instead of a successor of a limit cardinal –
Kunen’s Theorem excludes other cases), it is equal to $sup\{j^m(κ) : m
∈ ω\}$ where $j$ is the elementary embedding. Then $κ$ and $j^m(κ)$ are
$C^{(n)}$-$m$-huge (inter alia) in $V_δ$, for all $n$ and $m$.
If $κ$ is $C^{(n)}$-$\mathrm{I3}$, then it is $C^{(n)}$-$m$-huge, for
all $m$, and there is a normal ultrafilter $\mathcal{U}$ over $κ$ such
that
$\{α < κ : α$ is $C^{(n)}$-$m$-huge for every $m\} ∈ \mathcal{U}$.
## Consistency strength and size
Hugeness exhibits a phenomenon associated with similarly defined large
cardinals (the [](N-fold_variants.md))
known as the *double helix*. This phenomenon is when for one n-fold
variant, letting a cardinal be called n-$P_0$ iff it has the property,
and another variant, n-$P_1$, n-$P_0$ is weaker than n-$P_1$, which
is weaker than (n+1)-$P_0$. {% cite Kentaro2007 %} In
the consistency strength hierarchy, here is where these lay (top being
weakest):
- [measurable](Measurable.md "Measurable") =
0-[superstrong](Superstrong.md "Superstrong") =
0-huge
- n-superstrong
- n-fold supercompact
- (n+1)-fold strong, n-fold extendible
- (n+1)-fold Woodin, n-fold Vopěnka
- (n+1)-fold Shelah
- almost n-huge
- super almost n-huge
- n-huge
- super n-huge
- ultra n-huge
- (n+1)-superstrong
All huge variants lay at the top of the double helix restricted to some
[](Omega.md) n,
although each are bested by
<a href="Rank-into-rank" class="mw-redirect" title="Rank-into-rank">I3</a>
cardinals (the [](Elementary%20embedding.md)
of the I3 elementary embeddings). In fact, every I3 is preceeded by a
stationary set of n-huge cardinals, for all n.
{% cite Kanamori2009 %}
Similarly, every huge cardinal $\kappa$ is almost huge, and there is a
normal measure over $\kappa$ which contains every almost huge cardinal
$\lambda<\kappa$. Every superhuge cardinal $\kappa$ is
[extendible](Extendible.md "Extendible")
and there is a normal measure over $\kappa$ which contains every
extendible cardinal $\lambda<\kappa$. Every (n+1)-huge cardinal
$\kappa$ has a normal measure which contains every cardinal $\lambda$
such that $V_\kappa\modelsquot;$\lambda$ is super n-huge"
{% cite Kanamori2009 %}, in fact it contains every
cardinal $\lambda$ such that $V_\kappa\modelsquot;$\lambda$ is ultra
n-huge".
Every n-huge cardinal is m-huge for every m<n. Similarly with almost
n-hugeness, super n-hugeness, and super almost n-hugeness. Every almost
huge cardinal is
[Vopěnka](Vopenka.md "Vopenka")
(therefore the consistency of the existence of an almost-huge cardinal
implies the consistency of Vopěnka's principle).
{% cite Kanamori2009 %} Every ultra n-huge is super
n-huge and a stationary limit of super n-huge cardinals. Every super
almost (n+1)-huge is ultra n-huge and a stationary limit of ultra n-huge
cardinals.
In terms of size, however, the least n-huge cardinal is smaller than the
least
[supercompact](Supercompact.md "Supercompact")
cardinal (assuming both exist).
{% cite Kanamori2009 %} This is because n-huge
cardinals have upward reflection properties, while supercompacts have
downward reflection properties. Thus for any $\kappa$ which is
supercompact and has an n-huge cardinal above it, $\kappa$ "reflects
downward" that n-huge cardinal: there are $\kappa$-many n-huge
cardinals below $\kappa$. On the other hand, the least super n-huge
cardinals have *both* upward and downward reflection properties, and are
all *much* larger than the least supercompact cardinal. It is notable
that, while almost 2-huge cardinals have higher consistency strength
than superhuge cardinals, the least almost 2-huge is much smaller than
the least super almost huge.
While not every $n$-huge cardinal is
[strong](Strong.md "Strong"),
if $\kappa$ is almost $n$-huge with targets
$\lambda_1,\lambda_2...\lambda_n$, then $\kappa$ is
$\lambda_n$-strong as witnessed by the generated $j:V\prec M$. This
is because $j^n(\kappa)=\lambda_n$ is
[measurable](Measurable.md "Measurable")
and therefore $\beth_{\lambda_n}=\lambda_n$ and so
$V_{\lambda_n}=H_{\lambda_n}$ and because
$M^{<\lambda_n}\subset M$, $H_\theta\subset M$ for each
$\theta<\lambda_n$ and so
$\cup\{H_\theta:\theta<\lambda_n\} =
\cup\{V_\theta:\theta<\lambda_n\} = V_{\lambda_n}\subset
M$.
Every almost $n$-huge cardinal with targets
$\lambda_1,\lambda_2...\lambda_n$ is also
[$\theta$-supercompact](Supercompact.md "Supercompact")
for each $\theta<\lambda_n$, and every $n$-huge cardinal with
targets $\lambda_1,\lambda_2...\lambda_n$ is also
$\lambda_n$-supercompact.
For $2$-huge $κ$, $V_κ$ is a model of $\mathrm{ZFC}$+“there are proper
class many hyper-huge
cardinals”.{% cite Usuba2017 %} Hyper-huge
cardinals are extendible limits of extendible
cardinals.{% cite Usuba2019 %}
An $n$-huge* cardinal is an $n$-huge limit of $n$-huge cardinals. Every
$n + 1$-huge cardinal is
$n$-huge*.{% cite Gitmana %}
As for virtually
$n$-huge*:{% cite Gitmana %}
- If $κ$ is virtually huge*, then $V_κ$ is a model of proper class
many [](Extendible.md)
cardinals.
- A virtually $n+1$-huge* cardinal is a limit of virtually $n$-huge*
cardinals.
- A virtually $n$-huge* cardinal is an
$n+1$-<a href="Iterable" class="mw-redirect" title="Iterable">iterable</a>
limit of $n+1$-iterable cardinals. If $κ$ is $n+2$-iterable, then
$V_κ$ is a model of proper class many virtually $n$-huge*
cardinals.
- Every
<a href="Rank-into-rank" class="mw-redirect" title="Rank-into-rank">virtually rank-into-rank</a>
cardinal is a virtually $n$-huge* limit of virtually $n$-huge*
cardinals for every $n < ω$.
### The $\omega$-huge cardinals
A cardinal $\kappa$ is **almost $\omega$-huge** iff there is some
transitive model $M$ and an elementary embedding $j:V\prec M$ with
critical point $\kappa$ such that $M^{<\lambda}\subset M$ where
$\lambda$ is the smallest cardinal above $\kappa$ such that
$j(\lambda)=\lambda$. Similarly, $\kappa$ is **$\omega$-huge** iff
the model $M$ can be required to have $M^\lambda\subset M$.
Sadly, $\omega$-huge cardinals are inconsistent with ZFC by a version
of Kunen's inconsistency theorem. Now, $\omega$-hugeness is used to
describe critical points of
<a href="Rank-into-rank" class="mw-redirect" title="Rank-into-rank">I1 embeddings</a>.
## Relative consistency results
### Hugeness of $\omega_1$
In
<a href="https://projecteuclid.org/euclid.rmjm/1181073173" class="external autonumber">[2]</a>
it is shown that if $\text{ZFC +}$ "there is a huge cardinal" is
consistent then so is $\text{ZF +}$ "$\omega_1$ is a huge cardinal"
(with the ultrafilter characterization of hugeness).
### Generalizations of Chang's conjecture
### Cardinal arithmetic in $\text{ZF}$
If there is an almost huge cardinal then there is a model of
$\text{ZF+}\neg\text{AC}$ in which every successor cardinal is a
[Ramsey](Ramsey.md "Ramsey")
cardinal. It follows that (1) for all inner models $W$ of $\text{ZFC}$
and every singular cardinal $\kappa$, one has $\kappa^{+W} <
\kappa^+$ and that (2) for all ordinal $\alpha$ there is no injection
$\aleph_{\alpha+1}\to 2^{\aleph_\alpha}$. This in turn imply the
failure of the
<a href="index.php?title=Square_principle&action=edit&redlink=1" class="new" title="Square principle (page does not exist)">square principle</a>
at every infinite cardinal (and consequently
$\text{AD}^{L(\mathbb{R})}$, see
<a href="Determinacy" class="mw-redirect" title="Determinacy">determinacy</a>).
<a href="https://mathoverflow.net/questions/206090/what-consistency-results-follow-the-assumption-forall-alpha-aleph-alpha1" class="external autonumber">[3]</a>
## In set theoretic geology
If $\kappa$ is hyperhuge, then $V$ has lt;\kappa$ many
<a href="Ground" class="mw-redirect" title="Ground">grounds</a>
(so the
<a href="Mantle" class="mw-redirect" title="Mantle">mantle</a>
is a ground itself).{% cite Usuba2017 %} This
result has been strenghtened to
[extendible](Extendible.md "Extendible")
cardinals{% cite Usuba2019 %}. On
the other hand, it s consistent that there is a
[supercompact](Supercompact.md "Supercompact")
cardinal and class many grounds of $V$ (because of the indestructibility
properties of
supercompactness).{% cite Usuba2017 %}