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Superstrong cardinals were first utilized by Hugh Woodin in 1981 as an upper bound of consistency strength for the axiom of determinacy. However, Shelah had then discovered that Shelah cardinals were a weaker bound that still sufficed to imply the consistency strength of $\text{(ZF+)AD}$. After this, it was found that the existence of infinitely many Woodin cardinals was equiconsistent to $\text{AD}$. Woodin-ness is a significant weakening of superstrongness.

*Most results in this article can be found in
(Kanamori, 2009) unless indicated
otherwise.*

There are, like most critical point variations on measurable cardinals, multiple equivalent definitions of superstrongness. In particular, there is an elementary embedding definition and an extender definition.

A cardinal $\kappa$ is **$n$-superstrong** (or $n$-fold superstrong
when referring to the $n$-fold
variants)
iff it is the critical point of some elementary
embedding
$j:V\rightarrow M$ such that $M$ is a transitive class and
$V_{j^n(\kappa)}\subset M$ (in this case,
$j^{n+1}(\kappa):=j(j^n(\kappa))$ and $j^0(\kappa):=\kappa$).

A cardinal is **superstrong** iff it is $1$-superstrong.

The definition quite clearly shows that $\kappa$ is $j^n(\kappa)$-strong. However, the least superstrong cardinal is never strong.

A cardinal $\kappa$ is **$n$-superstrong** (or $n$-fold superstrong)
iff there is a
$(\kappa,\beta)$-extender
$\mathcal{E}$ for a $\beta>\kappa$ with
$V_{j^n_{\mathcal{E}}(\kappa)}\subseteq$
$Ult_{\mathcal{E}}(V)$
(where $j_{\mathcal{E}}$ is the canonical ultrapower embedding from
$V$ into $Ult_{\mathcal{E}}(V)$).

A cardinal is **superstrong** iff it is $1$-superstrong.

The consistency strength of $n$-superstrongness follows the double helix pattern (Kentaro, 2007). Specifically:

- measurable = $0$-superstrong = almost $0$-huge = super almost $0$-huge = $0$-huge = super $0$-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
- $(n+1)$-superstrong

Let $M$ be a transitive class $M$ such that there exists an elementary embedding $j:V\to M$ with $V_{j(\kappa)}\subseteq M$, and let $\kappa$ be its superstrong critical point. While $j(\kappa)$ need not be an inaccessible cardinal in $V$, it is always worldly and the rank model $V_{j(\kappa)}$ satisfies $\text{ZFC+}$”$\kappa$ is strong” (although $\kappa$ may not be strong in $V$).

Superstrong cardinals have strong upward reflection properties, in
particular there are many
measurable
cardinals *above* a superstrong cardinal. Every $n$-huge cardinal is
$n$-superstrong, and so $n$-huge cardinals also have strong reflection
properties. Remark however that if $\kappa$ is
strong
or
supercompact,
then it is consistent that there is no inaccessible cardinals larger
than $\kappa$: this is because if $\lambda>\kappa$ is
inaccessible, then $V_\lambda$ satisfies $\kappa$’s
strongness/supercompactness. Thus it is clear that supercompact
cardinals need not be superstrong, even though they have higher
consistency strength. In fact, because of the downward reflection
properties of strong/supercompact cardinals, if there is a superstrong
above a strong/supercompact $\kappa$, then there are $\kappa$-many
superstrong cardinals below $\kappa$; same with hugeness instead of
superstrongness. In particular, the least superstrong is strictly
smaller than the least strong (and thus smaller than the least
supercompact).

Every $1$-extendible cardinal is superstrong and has a normal measure containing all of the superstrongs less than said $1$-extendible. This means that the set of all superstronges less than it is stationary. Similarly, every cardinal $\kappa$ which is $2^\kappa$-supercompact is larger than the least superstrong cardinal and has a normal measure containing all of the superstrongs less than it.

Every superstrong cardinal is Woodin and has a normal measure containing all of the Woodin cardinals less than it. Thus the set of all Woodin cardinals below it is stationary, and so is the set of all measurables smaller than it. Superstrongness is consistency-wise stronger than Hyper-Woodinness.

If there is a superstrong cardinal, then in $L(\mathbb{R})$, the axiom of determinacy holds. (Jech, 2003)

Letting $\kappa$ be superstrong, $\kappa$ can be forced to $\aleph_2$ with an $\omega$-distributive, $\kappa$-c.c. notion of forcing, and in this forcing extension there is a normal $\omega_2$-saturated ideal on $\omega_1$. (Jech, 2003)

Superstrongness is not Laver indestructible. (Bagaria et al., 2013)

A cardinal $κ$ is
**$C^{(n)}$-superstrong**
iff there exists an elementary embedding $j : V → M$ for transitive $M$,
with $crit(j) = κ$, $V_{j(κ)} ⊆ M$ and $j(κ) ∈
C^{(n)}$.(Bagaria, 2012)

- Every $C^{(n)}$-superstrong cardinal belongs to $C^{(n)}$.
- Every superstrong cardinal is $C^{(1)}$-superstrong.
- For every $n ≥ 1$, if $κ$ is $C^{(n+1)}$-superstrong, then there is a $κ$-complete normal ultrafilter $U$ over $κ$ such that $\{α < κ : α$ is $C^{(n)}$-superstrong$\} ∈ U$. Hence, the first $C^{(n)}$-superstrong cardinal, if it exists, is not $C^{(n+1)}$-superstrong.
- If κ is $2^κ$-supercompact and belongs to $C^{(n)}$, then there is a $κ$-complete normal ultrafilter $U$ over $κ$ such that the set of $C^{(n)}$-superstrong cardinals smaller than $κ$ belongs to $U$.
- 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 $U$.
- 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 $C^{(n)}$-superstrong (inter alia) in $V_δ$, for all $n$ and $m$.

- 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 - Jech, T. J. (2003).
*Set Theory*(Third). Springer-Verlag. https://logic.wikischolars.columbia.edu/file/view/Jech%2C+T.+J.+%282003%29.+Set+Theory+%28The+3rd+millennium+ed.%29.pdf - 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 - Bagaria, J. (2012). \(C^{(n)}\)-cardinals.
*Archive for Mathematical Logic*,*51*(3–4), 213–240. https://doi.org/10.1007/s00153-011-0261-8