Climb into Cantor’s Attic, where you will find infinities large and small. We aim to provide a comprehensive resource of information about all notions of mathematical infinity.

View the Project on GitHub neugierde/cantors-attic

**Quick navigation**

The upper attic

The middle attic

The lower attic

The parlour

The playroom

The library

The cellar

**Sources**

Cantor's Attic (original site)

Joel David Hamkins blog post about the Attic

Latest working snapshot at the wayback machine

**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
on $\omega_1$”. It is now known that only a
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.
(Kanamori, 2009)

Their formulation is similar to that of the formulation of superstrong cardinals. A huge cardinal is to a supercompact cardinal as a superstrong cardinal is to a 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 as of modern set theoretic concerns. (Kentaro, 2007)

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.

The elementary embedding definitions are somewhat standard. Let $j:V\rightarrow M$ be a nontrivial elementary embedding of $V$ into a 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.

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 $\lambda\geq\kappa$.
[1]
Notice how similar this definition is to the alternative
characterization of
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.

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
$\lambda>\kappa$. (Usuba, 2017; Boney, 2017)

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 $j^n (κ) <
α$.(Gitman & Shindler, n.d.)

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 $j^n(κ)
< α$.(Gitman & Shindler, n.d.)

The first-order definition of n-huge is somewhat similar to measurability. Specifically, $\kappa$ is measurable iff there is a nonprincipal $\kappa$-complete ultrafilter, $U$, 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 of the poset $(X,\in)$. (Kanamori, 2009) $\kappa$ 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:(Boney, 2017)

- $κ$ 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.

(this section from (Bagaria, 2012))

$κ$ is
**$C^{(n)}$-$m$-huge**
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 ultrafilter $U$ 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 $\mathrm{I3}(κ, δ)$, 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 $m\} ∈ \mathcal{U}$.

Hugeness exhibits a phenomenon associated with similarly defined large
cardinals (the n-fold
variants)
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)-$P_0$. (Kentaro, 2007) In
the consistency strength hierarchy, here is where these lay (top being
weakest):

- measurable = 0-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 natural number n, although each are bested by I3 cardinals (the critical points of the I3 elementary embeddings). In fact, every I3 is preceeded by a stationary set of n-huge cardinals, for all n. (Kanamori, 2009)

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 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 $V_\kappa\models$”$\lambda$ is super n-huge” (Kanamori, 2009), in fact it contains every cardinal $\lambda$ such that $V_\kappa\models$”$\lambda$ 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 (therefore the consistency of the existence of an almost-huge cardinal implies the consistency of Vopěnka’s principle). (Kanamori, 2009) 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
cardinal (assuming both exist).
(Kanamori, 2009) 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, 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 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 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”.(Usuba, 2017) Hyper-huge cardinals are extendible limits of extendible cardinals.(Usuba, 2019)

An $n$-huge* cardinal is an $n$-huge limit of $n$-huge cardinals. Every $n + 1$-huge cardinal is $n$-huge*.(Gitman & Shindler, n.d.)

As for virtually $n$-huge*:(Gitman & Shindler, n.d.)

- If $κ$ is virtually huge*, then $V_κ$ is a model of proper class many virtually extendible cardinals.
- A virtually $n+1$-huge* cardinal is a limit of virtually $n$-huge* cardinals.
- A virtually $n$-huge* cardinal is an $n+1$-iterable 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 virtually rank-into-rank cardinal is a virtually $n$-huge* limit of virtually $n$-huge* cardinals for every $n < ω$.

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 I1 embeddings.

In [2] 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).

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 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 square principle at every infinite cardinal (and consequently $\text{AD}^{L(\mathbb{R})}$, see determinacy). [3]

If $\kappa$ is hyperhuge, then $V$ has $<\kappa$ many grounds (so the mantle is a ground itself).(Usuba, 2017) This result has been strenghtened to extendible cardinals(Usuba, 2019). On the other hand, it s consistent that there is a supercompact cardinal and class many grounds of $V$ (because of the indestructibility properties of supercompactness).(Usuba, 2017)

- Kanamori, A. (2009).
*The higher infinite*(Second, p. xxii+536). Springer-Verlag. https://link.springer.com/book/10.1007%2F978-3-540-88867-3 - Kentaro, S. (2007). Double helix in large large cardinals and iteration of elementary embeddings.
*Annals of Pure and Applied Logic*,*146*(2-3), 199–236. https://doi.org/10.1016/j.apal.2007.02.003 - Usuba, T. (2017). The downward directed grounds hypothesis and very large cardinals.
*Journal of Mathematical Logic*,*17*(02), 1750009. https://doi.org/10.1142/S021906131750009X - Boney, W. (2017).
*Model Theoretic Characterizations of Large Cardinals*. - Gitman, V., & Shindler, R.
*Virtual large cardinals*. https://ivv5hpp.uni-muenster.de/u/rds/virtualLargeCardinalsEdited5.pdf - Bagaria, J. (2012). \(C^{(n)}\)-cardinals.
*Archive for Mathematical Logic*,*51*(3–4), 213–240. https://doi.org/10.1007/s00153-011-0261-8 - Usuba, T. (2019). Extendible cardinals and the mantle.
*Archive for Mathematical Logic*,*58*(1-2), 71–75. https://doi.org/10.1007/s00153-018-0625-4