# Extendible cardinal

A cardinal $\kappa$ is $\eta$-extendible for an ordinal $\eta$ if and only if there is an elementary embedding $j:V_{\kappa+\eta}\to V_\theta$, with critical point $\kappa$, for some ordinal $\theta$. The cardinal $\kappa$ is extendible if and only if it is $\eta$-extendible for every ordinal $\eta$. Equivalently, for every ordinal $\alpha$ there is a nontrivial elementary embedding $j:V_{\kappa+\alpha+1}\to V_{j(\kappa)+j(\alpha)+1}$ with critical point $\kappa$.

## Alternative definition

Given cardinals $\lambda$ and $\theta$, a cardinal $\kappa\leq\lambda,\theta$ is jointly $\lambda$-supercompact and $\theta$-superstrong if there exists a nontrivial elementary embedding $j:V\to M$ for some transitive class $M$ such that $\mathrm{crit}(j)=\kappa$, $\lambda<j(\kappa)$, $M^\lambda\subseteq M$ and $V_{j(\theta)}\subseteq M$. That is, a single embedding witnesses both $\lambda$-supercompactness and (a strengthening of) superstrongness of $\kappa$. The least supercompact is never jointly $\lambda$-supercompact and $\theta$-superstrong for any $\lambda$,$\theta\geq\kappa$.

A cardinal is extendible if and only if it is jointly supercompact and $\kappa$-superstrong, i.e. for every $\lambda\geq\kappa$ it is jointly $\lambda$-supercompact and $\kappa$-superstrong. [1] One can show that extendibility of $\kappa$ is in fact equivalent to "for all $\lambda$,$\theta\geq\kappa$, $\kappa$ is jointly $\lambda$-supercompact and $\theta$-superstrong". A similar characterization of $C^{(n)}$-extendible cardinals exists.

The ultrahuge cardinals are defined in a way very similar to this, and one can (very informally) say that "ultrahuge cardinals are to superhuges what extendibles are to supercompacts". These cardinals are superhuge (and stationary limits of superhuges) and strictly below almost 2-huges in consistency strength.

To be expanded: Extendibility Laver Functions

## Relation to Other Large Cardinals

Extendible cardinals are related to various kinds of measurable cardinals.

Hyper-huge cardinals are extendible limits of extendible cardinals.[1]

The relationship between extendible, hypercompact and enhanced supercompact cardinals is not known. They all lay between supercompact and Vopěnka[2].

### Supercompactness

Extendibility is connected in strength with supercompactness. Every extendible cardinal is supercompact, since from the embeddings $j:V_\lambda\to V_\theta$ we may extract the induced supercompactness measures $X\in\mu\iff j''\delta\in j(X)$ for $X\subset \mathcal{P}_\kappa(\delta)$, provided that $j(\kappa)\gt\delta$ and $\mathcal{P}_\kappa(\delta)\subset V_\lambda$, which one can arrange. On the other hand, if $\kappa$ is $\theta$-supercompact, witnessed by $j:V\to M$, then $\kappa$ is $\delta$-extendible inside $M$, provided $\beth_\delta\leq\theta$, since the restricted elementary embedding $j\upharpoonright V_\delta:V_\delta\to j(V_\delta)=M_{j(\delta)}$ has size at most $\theta$ and is therefore in $M$, witnessing $\delta$-extendibility there.

Although extendibility itself is stronger and larger than supercompactness, $\eta$-supercompacteness is not necessarily too much weaker than $\eta$-extendibility. For example, if a cardinal $\kappa$ is $\beth_{\eta}(\kappa)$-supercompact (in this case, the same as $\beth_{\kappa+\eta}$-supercompact) for some $\eta<\kappa$, then there is a normal measure $U$ over $\kappa$ such that $\{\lambda<\kappa:\lambda\text{ is }\eta\text{-extendible}\}\in U$.

### Strong Compactness

Interestingly, extendibility is also related to strong compactness. A cardinal $\kappa$ is strongly compact iff the infinitary language $\mathcal{L}_{\kappa,\kappa}$ has the $\kappa$-compactness property. A cardinal $\kappa$ is extendible iff the infinitary language $\mathcal{L}_{\kappa,\kappa}^n$ (the infinitary language but with $(n+1)$-th order logic) has the $\kappa$-compactness property for every natural number $n$. [3]

Given a logic $\mathcal{L}$, the minimum cardinal $\kappa$ such that $\mathcal{L}$ satisfies the $\kappa$-compactness theorem is called the strong compactness cardinal of $\mathcal{L}$. The strong compactness cardinal of $\omega$-th order finitary logic (that is, the union of all $\mathcal{L}_{\omega,\omega}^n$ for natural $n$) is the least extendible cardinal.

## Variants

### $C^{(n)}$-extendible cardinals

(Information in this subsection from [4] unless noted otherwise)

A cardinal $κ$ is called $C^{(n)}$-extendible if for all $λ > κ$ it is $λ$-$C^{(n)}$-extendible, i.e. if there is an ordinal $µ$ and an elementary embedding $j : V_λ → V_µ$, with $\mathrm{crit(j)} = κ$, $j(κ) > λ$ and $j(κ) ∈ C^{(n)}$.

For $λ ∈ C^{(n)}$, a cardinal $κ$ is $λ$-$C^{(n)+}$-extendible iff it is $λ$-$C^{(n)}$-extendible, witnessed by some $j : V_λ → V_µ$ which (besides $j(κ) > λ$ and $j(κ) ∈ C(n)$) satisfies that $µ ∈ C^{(n)}$.

$κ$ is $C^{(n)+}$-extendible iff it is $λ$-$C^{(n)+}$-extendible for every $λ > κ$ such that $λ ∈ C^{(n)}$.

Properties:

• The notions of $C^{(n)}$-extendible cardinals and $C^{(n)+}$-extendible cardinals are equivalent.[5]
• Every extendible cardinal is $C^{(1)}$-extendible.
• If $κ$ is $C^{(n)}$-extendible, then $κ ∈ C^{(n+2)}$.
• For every $n ≥ 1$, if $κ$ is $C^{(n)}$-extendible and $κ+1$-$C^{(n+1)}$-extendible, then the set of $C^{(n)}$-extendible cardinals is unbounded below $κ$.
• Hence, the first $C^{(n)}$-extendible cardinal $κ$, if it exists, is not $κ+1$-$C^{(n+1)}$-extendible.
• In particular, the first extendible cardinal $κ$ is not $κ+1$-$C^{(2)}$-extendible.
• For every $n$, if there exists a $C^{(n+2)}$-extendible cardinal, then there exist a proper class of $C^{(n)}$-extendible cardinals.
• The existence of a $C^{(n+1)}$-extendible cardinal $κ$ (for $n ≥ 1$) does not imply the existence of a $C^{(n)}$-extendible cardinal greater than $κ$. For if $λ$ is such a cardinal, then $V_λ \models$“κ is $C^{(n+1)}$-extendible”.
• If $κ$ is $κ+1$-$C^{(n)}$-extendible and belongs to $C^{(n)}$, then $κ$ is $C^{(n)}$-superstrong and there is a $κ$-complete normal ultrafilter $U$ over $κ$ such that the set of $C^{(n)}$-superstrong cardinals smaller than $κ$ belongs to $U$.
• For $n ≥ 1$, the following are equivalent ($VP$ — Vopěnka's principle):
• $VP(Π_{n+1})$
• $VP(κ, \mathbf{Σ_{n+2}})$ for some $κ$
• There exists a $C(n)$-extendible cardinal.
• “For every $n$ there exists a $C(n)$-extendible cardinal.” is equivalent to the full Vopěnka's principle.
• Every $C^{(n)}$-superhuge cardinal is $C^{(n)}$-extendible.
• 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 $C^{(n)}$-extendible (inter alia) in $V_δ$, for all $n$ and $m$.

### $(\Sigma_n,\eta)$-extendible cardinals

There are some variants of extendible cardinals because of the interesting jump in consistency strength from $0$-extendible cardinals to $1$-extendibles. These variants specify the elementarity of the embedding.

A cardinal $\kappa$ is $(\Sigma_n,\eta)$-extendible, if there is a $\Sigma_n$-elementary embedding $j:V_{\kappa+\eta}\to V_\theta$ with critical point $\kappa$, for some ordinal $\theta$. These cardinals were introduced by Bagaria, Hamkins, Tsaprounis and Usuba [6].

### $\Sigma_n$-extendible cardinals

The special case of $\eta=0$ leads to a much weaker notion. Specifically, a cardinal $\kappa$ is $\Sigma_n$-extendible if it is $(\Sigma_n,0)$-extendible, or more simply, if $V_\kappa\prec_{\Sigma_n} V_\theta$ for some ordinal $\theta$. Note that this does not necessarily imply that $\kappa$ is inaccessible, and indeed the existence of $\Sigma_n$-extendible cardinals is provable in ZFC via the reflection theorem. For example, every $\Sigma_n$ correct cardinal is $\Sigma_n$-extendible, since from $V_\kappa\prec_{\Sigma_n} V$ and $V_\lambda\prec_{\Sigma_n} V$, where $\kappa\lt\lambda$, it follows that $V_\kappa\prec_{\Sigma_n} V_\lambda$. So in fact there is a closed unbounded class of $\Sigma_n$-extendible cardinals.

Similarly, every Mahlo cardinal $\kappa$ has a stationary set of inaccessible $\Sigma_n$-extendible cardinals $\gamma<\kappa$.

$\Sigma_3$-extendible cardinals cannot be Laver indestructible. Therefore $\Sigma_3$-correct, $\Sigma_3$-reflecting, $0$-extendible, (pseudo-)uplifting, weakly superstrong, strongly uplifting, superstrong, extendible, (almost) huge or rank-into-rank cardinals also cannot.[6]

### $A$-extendible cardinals

(this subsection from [7] unless noted otherwise)

Definitions:

• A cardinal $κ$ is $A$-extendible, for a class $A$, iff for every ordinal $λ > κ$ there is an ordinal $θ$ such that there is an elementary embedding
$j : ⟨ V_λ , ∈, A ∩ V_λ ⟩ → ⟨ V_θ , ∈, A ∩ V_θ ⟩$
with critical point $κ$ (such that $λ < j(κ)$ — removing this does not change, what cardinals are extendible).
• $λ$ is called the degree of $A$-extendibility of an embedding.
• A cardinal $κ$ is $(Σ_n)$-extendible, iff it is $A$-extendible, where $A$ is the $Σ_n$-truth predicate. (This is a different notion than $\Sigma_n$-extendible cardinals.)[5]
• For $A$-extendible $κ$ is, a set $g ⊆ κ$ is called $A$-stretchable, if for every $λ > κ$ and every $h ⊆ λ$ for which $h ∩ κ = g$, there is an elementary embedding $j : ⟨V_λ , ∈, A ∩ V_λ⟩ → ⟨V_θ , ∈, A ∩ V_θ⟩$ such that $crit(j)=κ$, $λ < j(κ)$ and $j(g) ∩ λ = h$.
• Intuitively, an $A$-stretchable set $g$ is one that can be stretched by an $A$-extendibility embedding to agree with any desired $h$ that extends $g$.
• $A$-strechability is a form of Laver diamond for $A$-extendibility.
• For $A$-extendible $κ$, a function $\ell : κ → V_κ$ is called an $A$-extendibility Laver function, if for every $λ$ and every target $a$, there is an elementary embedding $j : ⟨V_λ , ∈, A ∩ V_λ⟩ → ⟨V_θ , ∈, A ∩ V_θ⟩$ such that $crit(j)=κ$, $λ < j(κ)$ and $j(\ell)(κ) = a$.

Results:

• The following notions are equivalent:[5]
• $C^{(n)}$-extendibility in the sense of Bagaria (defined in a section above)
• $A$-extendibility where $A$ is the class $C^{(n)}$
• $(Σ_n)$-extendibility
• $κ$ is $A$-extendible for every $Σ_n$-definable class $A$, allowing parameters in $V_κ$
• The Vopěnka principle is equivalent over GBC to both following statements:
• For every class $A$, there is an $A$-extendible cardinal.
• For every class $A$, there is a stationary proper class of $A$-extendible cardinals.
• If $κ$ is $A$-extendible for some class $A$, then
• there is an $A$-stretchable set $g ⊆ κ$.
• there is an $A$-extendibility Laver function $\ell : κ → V_κ$.
• In $\text{GBC}$, for any class $A$ there is a class function $\ell : \mathrm{Ord} → V$, such that for every $A$-extendible cardinal $κ$, $\ell ↾ κ$ is an $A$-extendible Laver function for $κ$.
• This uses global well-ordering that is a consequence of global choice.
• Without global choice, one can still have ordinal-anticipating Laver function $\ell : \mathrm{Ord} → \mathrm{Ord}$ and get for example $A$-extendibility Menas property.
• If $κ$ is $A$-extendible for a class $A$, then $κ$ is $Σ_2(A)$-reflecting.
• If $κ$ is $A ⊕ C$-extendible, where $C$ is the class of all $Σ_1(A)$-correct ordinals, then $κ$ is $Σ_3(A)$-reflecting.

### Virtually extendible cardinals

Definitions:

• A cardinal $κ$ is (weakly? strongly? ......) virtually extendible iff for every $α > κ$, in a set-forcing extension there is an elementary embedding $j : V_α → V_β$ with $\mathrm{crit(j)} = κ$ and $j(κ) > α$.
• $C^{(n)}$-virtually extendible cardinals require additionally that $j(κ)$ has property $C^{(n)}$ (i.e. $\Sigma_n$-correctness).[8]
• A cardinal $κ$ is (weakly) virtually $A$-extendible, for a class $A$, iff for every ordinal $λ > κ$ there is an ordinal $θ$ such that in a set-forcing extension, there is an elementary embedding
$j : \langle V_λ , ∈, A ∩ V_λ \rangle → \langle V_θ , ∈, A ∩ V_θ \rangle$
with critical point $κ$.
• For (strongly) virtually $A$-extendible $κ$, we require additionally $λ < j(κ)$.[5]
• A cardinal $κ$ is $n$-remarkable, for $n > 0$, iff for every $η > κ$ in $C^{(n)}$ , there is $α<κ$ also in $C^{(n)}$ such that in $V^{Coll(ω, < κ)}$, there is an elementary embedding $j : V_α → V_η$ with $j(\mathrm{crit}(j)) = κ$.
• A cardinal is completely remarkable iff it is $n$-remarkable for all $n > 0$.[9]
• A cardinal κ is weakly or strongly virtually $(Σ_n)$-extendible, iff it is respectively weakly or strongly virtually $A$-extendible, where $A$ is the $Σ_n$-truth predicate.[5]

Equivalence and hierarchy:

• $1$-remarkability is equivalent to remarkability. A cardinal is virtually $C^{(n)}$-extendible iff it is $n + 1$-remarkable (virtually extendible cardinals are virtually $C^{(1)}$-extendible).[9]
• Weakly and strongly $A$-extendible cardinal are non-equivalent, although in the non-virtual context, the weak and strong forms of $A$-extendibility coincide.[5]
• It is relatively consistent with GBC that every class $A$ admits a (weakly) virtually $A$-extendible cardinal (and so the generic Vopěnka principle holds), but no class $A$ admits a (strongly) virtually $A$-extendible cardinal.[5]
• Every $n$-remarkable cardinal is in $C^{(n+1)}$.[9]
• Every $n+1$-remarkable cardinal is a limit of $n$-remarkable cardinals.[9]

Upper limits for strength:

• If $κ$ is virtually Shelah for supercompactness or 2-iterable, then $V_κ$ is a model of proper class many virtually $C^{(n)}$-extendible cardinals for every $n < ω$.[8]
• If $κ$ is virtually huge*, then $V_κ$ is a model of proper class many virtually extendible cardinals.[8]
• Completely remarkable cardinals can exist in $L$.[9]
• For a $2$-iterable cardinal $κ$, $V_κ$ is a model of proper class many completely remarkable cardinals.[9]
• If $0^\#$ exists, then every Silver indiscernible is in $L$ completely remarkable and virtually $A$-extendible for every definable class $A$.[5, 9]

Lower limit for strength:

• Virtually extendible cardinals are remarkable limits of remarkable cardinals and 1-iterable limits of 1-iterable cardinals.[8]

In relation to Generic Vopěnka's Principle:(from [9] unless noted otherwise)

• The following are equiconsistent
• $gVP(Π_n)$
• $gVP(κ, \mathbf{Σ_{n+1}})$ for some $κ$
• There is an $n$-remarkable cardinal.
• The following are equiconsistent
• $gVP(\mathbf{Π_n})$
• $gVP(κ, \mathbf{Σ_{n+1}})$ for a proper class of $κ$
• There is a proper class of $n$-remarkable cardinals.
• Unless there is a transitive model of ZFC with a proper class of $n$-remarkable cardinals,
• if for some cardinal $κ$, $gVP(κ, \mathbf{Σ_{n+1}})$ holds, then there is an $n$-remarkable cardinal.
• if $gVP(Π_n)$ holds, then there is an $n$-remarkable cardinal.
• if $gVP(\mathbf{Π_n})$ holds, then there is a proper class of $n$-remarkable cardinals.
• $κ$ is the least for which $gVP^∗(κ, \mathbf{Σ_{n+1}})$ holds. $\iff κ$ is the least $n$-remarkable cardinal.
• If $gVP^∗(Π_n)$ holds, then there is an $n$-remarkable cardinal.
• If $gVP^∗(\mathbf{Π_n})$ holds, then there is a proper class of $n$-remarkable cardinals.
• If there is a proper class of $n$-remarkable cardinals, then $gVP(Σ_{n+1})$ holds.[5]
• If $gVP(Σ_{n+1})$ holds, then either there is a proper class of $n$-remarkable cardinals or there is a proper class of virtually rank-into-rank cardinals.[5]
• The generic Vopěnka principle holds iff for every class $A$, there are a proper class of (weakly) virtually $A$-extendible cardinals.[5]
• The generic Vopěnka scheme is equivalent over ZFC to the scheme asserting of every definable class $A$ that there is a proper class of weakly virtually $A$-extendible cardinals.[5]
• For $n ≥ 1$, the following are equivalent as schemes over ZFC:[5]
• The generic Vopěnka scheme holds for $Π_{n+1}$-definable classes.
• The generic Vopěnka scheme holds for $Σ_{n+2}$-definable classes.
• For every $Σ_n$-definable class A, there is a proper class of (weakly) virtually $A$-extendible cardinals.
• There is a proper class of (weakly) virtually $(Σ_n)$-extendible cardinals.
• There is a proper class of cardinals $κ$, such that for every $Σ_n$-correct cardinal $λ>κ$, there is a $Σ_n$-correct cardinal $θ > λ$ and a virtual elementary embedding $j : V_λ → V_θ$ with $crit(j)=κ$.
• If $0^♯$ exists, then there is a class-forcing extension $L[G]$ of the constructible universe in which the generic Vopěnka principle holds (so $gVP(κ, \mathbf{Σ_{n+1}})$ and $gVP(Π_n)$ hold for any $κ$ and $n$), but there are no $Σ_2$-reflecting cardinals and hence no remarkable cardinals (or $n$-remarkable cardinals).[5]

## In set-theoretic geology

If $κ$ is extendible then the $κ$-mantle of $V$ is its smallest ground (so of course the mantle is a ground of V).[1]

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## References

1. Usuba, Toshimichi. Extendible cardinals and the mantle. Archive for Mathematical Logic 58(1-2):71-75, 2019. arχiv   DOI   bibtex
2. Perlmutter, Norman. The large cardinals between supercompact and almost-huge. , 2010. www   arχiv   bibtex
3. 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
4. Bagaria, Joan. $C^{(n)}$-cardinals. Archive for Mathematical Logic 51(3--4):213--240, 2012. www   arχiv   DOI   bibtex
5. Gitman, Victoria and Hamkins, Joel David. A model of the generic Vopěnka principle in which the ordinals are not Mahlo. , 2018. arχiv   bibtex
6. Bagaria, Joan and Hamkins, Joel David and Tsaprounis, Konstantinos and Usuba, Toshimichi. Superstrong and other large cardinals are never Laver indestructible. Archive for Mathematical Logic 55(1-2):19--35, 2013. www   arχiv   DOI   bibtex
7. Hamkins, Joel David. The Vopěnka principle is inequivalent to but conservative over the Vopěnka scheme. , 2016. www   arχiv   bibtex
8. Gitman, Victoria and Shindler, Ralf. Virtual large cardinals. www   bibtex
9. Bagaria, Joan and Gitman, Victoria and Schindler, Ralf. Generic {V}opěnka's {P}rinciple, remarkable cardinals, and the weak {P}roper {F}orcing {A}xiom. Arch Math Logic 56(1-2):1--20, 2017. www   DOI   MR   bibtex
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