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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 $\kappa$.
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
Extendible cardinals are related to various kinds of measurable cardinals.
Hyper-huge cardinals are extendible limits of extendible cardinals.(Usuba, 2019)
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 $\{\lambda<\kappa:\lambda\text{ is }\eta\text{-extendible}\}\in U$.
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 $n$. (Kanamori, 2009)
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.
(Information in this subsection from (Bagaria, 2012) 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:
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 (Bagaria et al., 2013).
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.(Bagaria et al., 2013)
(this subsection from (Hamkins, 2016))
Definitions:
Results:
Definitions:
Equivalence and hierarchy:
Upper limits for strength:
Lower limit for strength:
In relation to Generic Vopěnka’s Principle:(from (Bagaria et al., 2017) unless noted otherwise)
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If $κ$ is extendible then the $κ$-mantle of $V$ is its smallest ground (so of course the mantle is a ground of V).(Usuba, 2019)
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