Huge cardinal
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.[1]
Contents
Definitions
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. [2]
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.
Elementary embedding definition
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.
Note: $M^{<\lambda}$ and $M^{\lambda}$ can also be denoted $^{<\lambda}M$ and $^{\lambda}M$.
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 $\lambda\geq\kappa$. [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.
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 $\lambda>\kappa$.[3, 4]
$\lambda$-hyperhuge and hyperhuge are other names for $2$-fold $\lambda$-supercompact and $2$-fold supercompact.
Huge* cardinals
A cardinal $\kappa$ is $n$-huge* if for some $\alpha\gt\kappa$, $\kappa$ is the critical point of an elementary embedding $j: V_\alpha\prec V_\beta$ such that $j^n(\kappa)\lt\alpha$.[5]
Hugeness* variant is formulated in a way allowing for a virtual variant consistent with $V=L$: A cardinal $\kappa$ is virtually $n$-huge* if for some $\alpha\gt\kappa$, in a set-forcing extension, $\kappa$ is the critical point of an elementary embedding $j: V_\alpha\prec V_\beta$ such that $j^n(\kappa)\lt\alpha$.[5]
Ultrafilter definition
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 $(X,\in)$. [1] $\kappa$ 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:[4]
- $\kappa$ is $\lambda$-hyperhuge;
- $\mu\lt\lambda$ and a normal, fine, $\kappa$-complete ultrafilter exists on $[\mu]^\lambda_{∗\kappa} := \{s\subseteq μ:|s|=\lambda, |s\cap\kappa|\lt\kappa, \mathrm{otp}(s\cap\lambda)\lt\kappa\}$;
- $\mathbb{L}_{\kappa,\kappa}$ is $[\mu]^\lambda_{∗\kappa}$-$\kappa$-compact for type omission.
Coherent sequence characterization of almost hugeness
Almost huge cardinals can be characterized via coherent sequences of ultrafilters, so that the direct limit of the ultrapowers witnesses the almost hugeness of $\kappa$ and each individual ultrapower witnesses a $j_\gamma: V\prec M_\gamma$ with $\gamma\lt j(\kappa)$ and $M^\gamma\subseteq M$. $\kappa$ is almost-huge if there is some inaccessible $\lambda\gt\kappa$ and a coherent sequence of normal ultrafilters $\langle \mathcal{U}_\gamma|\kappa\le\gamma\lt\lambda\rangle$ over $\mathcal{P}_\kappa(\gamma)$ such that the corresponding embeddings $j_\gamma : V\to M_\gamma\cong Ult(V, \mathcal{U}_\gamma)$ and $k_{\gamma,\delta}: M_\gamma\to M_\delta$ satisfy: if $\kappa\le\gamma\lt\lambda$ and $\gamma\le\alpha\lt j_\gamma(\kappa)$, then there is $\delta$ such that $\gamma\le\delta\lt\lambda$ and $k_{\gamma,\delta}(\alpha)=\delta$.[1]
Because each $\mathcal{U}_\gamma$ is a normal fine ultrafilter, $M_\gamma^\gamma\subseteq M_\gamma$ and $j_\gamma(\kappa)\gt\gamma$. Also, if $\hat M$ is the direct limit and $\hat j: V\prec\hat M$, then for every $\beta\lt\hat j(\kappa)$, $M^\beta\subseteq\beta$ and $\text{crit}j=\kappa$. $j=k_\gamma\circ j_\gamma$ for $k_\gamma$ the factor embedding $k_\gamma: M_\gamma\prec\hat M$, and $\lambda=\hat j(\kappa)$.
$C^{(n)}$-$m$-huge cardinals
(this section from [6], information about almost-huge and superhuge (and other, if noted) from 2019 extended arXiv version)
Elementary-embedding definitions:
- $κ$ is $C^{(n)}$-$m$-huge iff it is $m$-huge and $j(κ) ∈ C^{(n)}$ ($C^{(n)}$-huge if it is huge and $j(κ) ∈ C^{(n)}$ ($m=1$)).
- $κ$ is $C^{(n)}$-$m$-almost-huge iff it is almost-huge and $j(κ) ∈ C^{(n)}$.
- $κ$ is $C^{(n)}$-superhuge iff it is superhuge and there are witnessing embeddings with arbitrarily large $j(κ) ∈ C^{(n)}$.
Equivalent definitions in terms of normal measures:
- $κ$ is $C^{(n)}$-$m$-huge iff it is uncountable and there is a $κ$-complete normal ultrafilter $\mathcal{U}$ over some $\mathcal{P}(λ)$ and cardinals $κ = λ_0 < λ_1 < . . . < λ_m = λ$, with $λ_1 ∈ C^{(n)}$ and such that for each $i < m$, $\{x ∈ \mathcal{P}(λ) : ot(x ∩ λ_{i+1}) = λ_i \} ∈ \mathcal{U}$.
- $κ$ is $C^{(n)}$-almost-huge iff there is an inaccessible $λ ∈ C^{(n)}$ greater than $κ$ and a coherent sequence of normal ultrafilters $\langle \mathcal{U}_γ : κ ≤ γ < λ \rangle$ over $\mathcal{P}_κ(γ)$ such that the corresponding embeddings $j_γ : V → M_γ ≅ Ult(V, \mathcal{U}_γ )$ and $k_{γ,δ} : M_γ → M_δ$ satisfy: if $κ ≤ γ < λ$ and $γ ≤ α < j_γ (κ)$, then there is $δ$ such that $γ ≤ δ < λ$ and $k_{γ,δ}(α) = δ$. (See [1], 24.11 in 1994 edition for details.)
- $κ$ is $C^{(n)}$-superhuge iff for every $α$ there is a $κ$-complete fine and normal ultrafilter $\mathcal{U}$ over some $\mathcal{P}(λ)$, with $λ ∈ C^{(n)}$ and such that $\{x ∈ \mathcal{P}(λ) : ot(x) = κ \} ∈ \mathcal{U}$.
It follows that
- “$κ$ is $C^{(n)}$-$m$-huge” and “$κ$ is $C^{(n)}$-almost-huge” are $Σ_{n+1}$ expressible and
- “$κ$ is $C^{(n)}$-superhuge” is $Π_{n+2}$ expressible.
Every huge cardinal is $C^{(1)}$-huge. Every superhuge cardinal is $C^{(1)}$-superhuge.
Hierarchy:
- 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.
- Similar argumentation shows that he first $C^{(n)}$-superhuge cardinal is not $C^{(n+1)}$-superhuge.
- If $κ$ is $C^{(n)}$-$2$-huge, then there is a $κ$-complete normal ultrafilter $\mathcal{U}$ over $κ$ such that $\{α < κ : V_κ \models$ “$α$ is $C^{(n)}$-superhuge” $\} ∈ \mathcal{U}$.
- If $κ$ is $C^{(n)}$-huge, then it is $C^{(n)}$-almost-huge and there is a $κ$-complete normal ultrafilter $\mathcal{U}$ over $κ$ such that $\{α < κ : α$ is $C^{(n)}$-almost-huge$\} ∈ \mathcal{U}$
- Every $C^{(n)}$-almost-huge cardinal is $C^{(n)}$-superstrong, so it belongs to $C^{(n)}$.
- Therefore, taking into account that $C^{(n)}$-huge is $Σ_{n+1}$ expressible, the first $C^{(n)}$-huge cardinal is smaller than the first $C^{(n+1)}$-almost-huge cardinal (provided both exist).
- Every $C^{(n)}$-almost-huge cardinal is $C^{(n)}$-superstrong, so it belongs to $C^{(n)}$.
Relations with other large cardinals:
- If $κ$ is $C^{(n)}$-superhuge, then $κ$ is $C^{(n)}$-extendible (in particular $κ ∈ C^{(n+2)}$) and there is a $κ$-complete normal ultrafilter $\mathcal{U}$ over $κ$ such that $\{α < κ : α$ is $C^{(n)}$-extendible$\} ∈ \mathcal{U}$. $κ$ is also $C^{(n)}$-supercompact.
- 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 and $C^{(n)}$-superhuge (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 ($κ$-complete — from 2019 version) 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) 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)-$P_0$. [2] 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. [1]
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" [1], in fact it contains every cardinal $\lambda$ such that $V_\kappa\models$"$\lambda$ is ultra n-huge". In fact, if $\kappa$ is 2-huge, it contains $V_\kappa\vDash\lambda\text{ is hyperhyge}$
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). [1] 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). [1] 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.
If $n\ge 1$ and $\mathrm{I}_4^{n+1}(\kappa)$, then $\kappa$ is $n$-huge and $\{\alpha\lt\kappa|\alpha\text{ is }n\text{-huge}\}$ has measure 1.[7]
For $2$-huge $κ$, $V_κ$ is a model of $\mathrm{ZFC}$+“there are proper class many hyper-huge cardinals”.[3] Hyper-huge cardinals are extendible limits of extendible cardinals.[8]
An $n$-huge* cardinal is an $n$-huge limit of $n$-huge cardinals. Every $n + 1$-huge cardinal is $n$-huge*.[5]
As for virtually $n$-huge*:[5]
- 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 $n < ω$.
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 I1 embeddings.
Relative consistency results
Hugeness of $\omega_1$
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).
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 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]
In set theoretic geology
If $\kappa$ is hyperhuge, then $V$ has $<\kappa$ many grounds (so the mantle is a ground itself).[3] This result has been strenghtened to extendible cardinals[8]. 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).[3]
References
- Kanamori, Akihiro. The higher infinite. Second, Springer-Verlag, Berlin, 2009. (Large cardinals in set theory from their beginnings, Paperback reprint of the 2003 edition) www bibtex
- Kentaro, Sato. Double helix in large large cardinals and iteration of elementary embeddings. Annals of Pure and Applied Logic 146(2-3):199-236, May, 2007. www DOI bibtex
- Usuba, Toshimichi. The downward directed grounds hypothesis and very large cardinals. Journal of Mathematical Logic 17(02):1750009, 2017. arχiv DOI bibtex
- Boney, Will. Model Theoretic Characterizations of Large Cardinals. arχiv bibtex
- Gitman, Victoria and Shindler, Ralf. Virtual large cardinals. www bibtex
- Bagaria, Joan. $C^{(n)}$-cardinals. Archive for Mathematical Logic 51(3--4):213--240, 2012. www arχiv DOI bibtex
- Corazza, Paul. The gap between ${\rm I}_3$ and the wholeness axiom. Fund Math 179(1):43--60, 2003. www DOI MR bibtex
- Usuba, Toshimichi. Extendible cardinals and the mantle. Archive for Mathematical Logic 58(1-2):71-75, 2019. arχiv DOI bibtex