Extendible cardinal

• Alternative definition
• Relation to Other Large Cardinals
  • Supercompactness
  • Strong Compactness
• Variants
  • $C^{(n)}$-extendible cardinals
  • $(\Sigma_n,\eta)$-extendible cardinals
  • $\Sigma_n$-extendible cardinals
  • $A$-extendible cardinals
  • Virtually extendible cardinals
• In set-theoretic geology

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.(Usuba, 2019)

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 $n$. (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.

Variants

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

(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:

• The notions of $C^{(n)}$-extendible cardinals and $C^{(n)+}$-extendible cardinals are equivalent.(Gitman & Hamkins, 2018)
• 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.
• 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 (Bagaria et al., 2013).

$\Sigma_n$-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.(Bagaria et al., 2013)

$A$-extendible cardinals

(this subsection from (Hamkins, 2016))

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 $j : \langle V_λ , ∈, A ∩ V_λ \rangle → \langle V_θ , ∈, A ∩ V_θ \rangle$ 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.)(Gitman & Hamkins, 2018)

Results:

• 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.
• ……

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).(Gitman & Shindler, n.d.)
• 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 $λ < j(κ)$.(Gitman & Hamkins, 2018)
• 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 $n > 0$.(Bagaria et al., 2017)
• 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.(Gitman & Hamkins, 2018)

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).(Bagaria et al., 2017)
• Weakly and strongly $A$-extendible cardinal are non-equivalent, although in the non-virtual context, the weak and strong forms of $A$-extendibility coincide.(Gitman & Hamkins, 2018)
• 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.(Gitman & Hamkins, 2018)
• Every $n$-remarkable cardinal is in $C^{(n+1)}$.(Bagaria et al., 2017)
• Every $n+1$-remarkable cardinal is a limit of $n$-remarkable cardinals.(Bagaria et al., 2017)

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 $n < ω$.(Gitman & Shindler, n.d.)
• If $κ$ is virtually huge*, then $V_κ$ is a model of proper class many virtually extendible cardinals.(Gitman & Shindler, n.d.)
• Completely remarkable cardinals can exist in $L$.(Bagaria et al., 2017)
• For a $2$-iterable cardinal $κ$, $V_κ$ is a model of proper class many completely remarkable cardinals.(Bagaria et al., 2017)
• If $0^\#$ exists, then every Silver indiscernible is in $L$ completely remarkable and virtually $A$-extendible for every definable class $A$. (Gitman & Hamkins, 2018; Bagaria et al., 2017)

Lower limit for strength:

• Virtually extendible cardinals are remarkable limits of remarkable cardinals and 1-iterable limits of 1-iterable cardinals.(Gitman & Shindler, n.d.)

In relation to Generic Vopěnka’s Principle:(from (Bagaria et al., 2017) 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.
• $κ$ is the least for which $gVP^∗(κ, \mathbf{Σ_{n+1}})$ holds. $\iff κ$ is the least $n$-remarkable cardinal.
• If $gVP^∗(Π_n)$, 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.(Gitman & Hamkins, 2018)
• 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.(Gitman & Hamkins, 2018)
• 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.(Gitman & Hamkins, 2018)
• Open problems: Must there be an $n$-remarkable cardinal
  • if $gVP(κ, \mathbf{Σ_{n+1}})$ holds for some $κ$.
  • if $gVP(Π_n)$ holds.

……

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).(Usuba, 2019)

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References

1. 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
2. Kanamori, A. (2009). The higher infinite (Second, p. xxii+536). Springer-Verlag. https://link.springer.com/book/10.1007%2F978-3-540-88867-3
3. Bagaria, J. (2012). \(C^{(n)}\)-cardinals. Archive for Mathematical Logic, 51(3–4), 213–240. https://doi.org/10.1007/s00153-011-0261-8
4. Gitman, V., & Hamkins, J. D. (2018). A model of the generic Vopěnka principle in which the ordinals are not Mahlo.
5. Bagaria, J., Hamkins, J. D., Tsaprounis, K., & Usuba, T. (2013). Superstrong and other large cardinals are never Laver indestructible. Archive for Mathematical Logic, 55(1-2), 19–35. https://doi.org/10.1007/s00153-015-0458-3
6. Hamkins, J. D. (2016). The Vopěnka principle is inequivalent to but conservative over the Vopěnka scheme. http://jdh.hamkins.org/vopenka-principle-vopenka-scheme/
7. Gitman, V., & Shindler, R. Virtual large cardinals. https://ivv5hpp.uni-muenster.de/u/rds/virtualLargeCardinalsEdited5.pdf
8. Bagaria, J., Gitman, V., & Schindler, R. (2017). Generic Vopěnkaś Principle, remarkable cardinals, and the weak Proper Forcing Axiom. Arch. Math. Logic, 56(1-2), 1–20. https://doi.org/10.1007/s00153-016-0511-x

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