Difference between revisions of "Superstrong"

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A cardinal $\kappa$ is ''n-superstrong'' if there is an elementary embedding $j:V\to M$ of the universe $V$ into a transitive class $M$, with critical point $\kappa$, such that $V_{j^n(\kappa)}\subset M$.  
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[[Category:Large cardinal axioms]]
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[[Category:Critical points]]
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Superstrong cardinals were first utilized by Hugh Woodin in 1981 as an upper bound of consistency strength for [[AD]]. However, Shelah had then discovered that [[Shelah]] cardinals were a weaker bound that still sufficed to imply the consistency strength for [[AD]]. After this, it was found that the existence of a proper class of [[Woodin]] cardinals was equiconsistent to [[AD]]. This is a significant weakening of superstrongness.
  
$\kappa$ is ''superstrong'' if it is 1-superstrong. 0-superstrongness is equivalent to [[measurable|measurability]].
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== Definitions ==
  
=== Relation to other large cardinal notions ===
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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.
  
Every superstrong cardinal is a [[Shelah | Shelah cardinal]], and a [[Woodin | Woodin cardinal]], also there are $\kappa$ Woodin cardinals below it. Every 1-[[extendible]] cardinal is superstrong; if $\kappa$ is $2^\kappa$-[[supercompact]] or 1-extendible then there are $\kappa$ superstrong cardinals below $\kappa$.
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=== Elementary Embedding Definition ===
  
If $\kappa$ is superstrong as witnessed by $j:V \to M$, then $\kappa$ is a [[strong | strong cardinal]] in the model $M_{j(\kappa)}=M\cap V_{j(\kappa)}$.
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A cardinal $\kappa$ is '''$n$-superstrong''' (or $n$-fold superstrong when referring to the [[n-fold variants|$n$-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 $j^0(\kappa):=\kappa$).
However, superstrong cardinals need not be strong in $V$.
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In fact, the least superstrong cardinal is less than the least strong cardinal, because superstrongness is a $\Sigma_2$ property (as can be seen using its characterization in terms of [[extender | extenders]])
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whereas every strong cardinal is $\Sigma_2$-[[reflecting]]. Furthermore, if $\kappa$ is strong and there is a superstrong cardinal above it, then there are $\kappa$ superstrong cardinals below $\kappa$.
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The existence of a (n+1)-superstrong cardinal implies the consistency of the existence of a n-superstrong cardinal, furthermore it also implies the consistency of the existence of a [[huge|n-huge]] cardinal (for n > 0).
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A cardinal is '''superstrong''' iff it is $1$-superstrong.
  
=== References ===
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The definition quite clearly shows that $\kappa$ is [[strong|$j^n(\kappa)$-strong]]. However, the least superstrong cardinal is never [[strong]].
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<cite>Kanamori2009:HigherInfinite</cite>
  
Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nd ed.). Springer. ISBN 3-540-00384-3.
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=== Extender Definition ===
  
{{stub}}
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A cardinal $\kappa$ is '''$n$-superstrong''' (or $n$-fold superstrong when referring to the [[n-fold variants|$n$-fold variants]]) iff there is a [[extender|$(\kappa,\beta)$-extender]] $\mathcal{E}$ for a $\beta>\kappa$ with $V_{j^n_{\mathcal{E}}(\kappa)}\subseteq$ [[ultrapower|$Ult_{\mathcal{E}}(V)$]] (where $j_{\mathcal{E}}$ is the canonical ultrapower embedding from $V$ into $Ult_{\mathcal{E}}(V)$). <cite>Kanamori2009:HigherInfinite</cite>
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A cardinal is '''superstrong''' iff it is $1$-superstrong.
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== Relation to other large cardinal notions ==
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*The consistency strength of $n$-superstrongness follows the [[n-fold variants|double helix pattern]]. <cite>Kentaro2007:DoubleHelix</cite> Specifically:
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**[[measurable]] = $0$-superstrong = [[huge|almost $0$-huge]] = super almost $0$-huge = $0$-huge = super $0$-huge
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**$n$-superstrong
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**[[n-fold variants|$n$-fold supercompact]]
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**[[n-fold variants|$(n+1)$-fold strong]], [[n-fold variants|$n$-fold extendible]]
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**[[n-fold variants|$(n+1)$-fold Woodin]], [[n-fold variants|$n$-fold Vopěnka]]
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**[[n-fold variants|$(n+1)$-fold Shelah]]
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**almost $n$-huge
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**super almost $n$-huge
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**$n$-huge
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**super $n$-huge
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**$(n+1)$-superstrong
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*The consistency strength of the existence of a superstrong cardinal is less than that of the existence of a $\kappa$ which is [[supercompact|$2^\kappa$-supercompact]].  <cite>Kanamori2009:HigherInfinite</cite>
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*If $\kappa$ is superstrong, then the necessary transitive class $M$ has $M\cap V_{j(\kappa)}$ satisfying $\kappa$'s [[strong|strongness]] (although $\kappa$ may not be strong in $V$).
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*There are many [[measurable]] cardinals above a superstrong cardinal. <cite>Kanamori2009:HigherInfinite</cite>
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*Every $n$-huge cardinal is $n$-superstrong. <cite>Kanamori2009:HigherInfinite</cite>
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*If $\kappa$ is strong but there is a superstrong cardinal at least the size of $\kappa$, then there are also $\kappa$ superstrong cardinals less than $\kappa$. <cite>Kanamori2009:HigherInfinite</cite>
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*Every [[extendible|$1$-extendible]] cardinal is superstrong and has a [[filter|normal measure]] containing all of the superstrongs less than said $1$-extendible. <cite>Kanamori2009:HigherInfinite</cite>
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*Every $\kappa$ which is [[supercompact|$2^\kappa$-supercompact]] is larger than a superstrong cardinal and has a normal measure containing all of the superstrongs less than it. <cite>Kanamori2009:HigherInfinite</cite>
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*Every superstrong cardinal is [[Woodin]] and has a normal measure containing all of the Woodin cardinals less than it. <cite>Kanamori2009:HigherInfinite</cite>
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*Superstrongness is consistency-wise stronger than [[Woodin|Hyper-Woodinness]]. <cite>Kanamori2009:HigherInfinite</cite>
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*Superstrongness is not Laver indestructible. <cite>BagariaHamkinsTsaprounisUsuba:SuperstrongAndOtherLargeCardinalsAreNeverLaverIndestructible</cite>
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{{References}}

Revision as of 00:29, 8 November 2017

Superstrong cardinals were first utilized by Hugh Woodin in 1981 as an upper bound of consistency strength for AD. However, Shelah had then discovered that Shelah cardinals were a weaker bound that still sufficed to imply the consistency strength for AD. After this, it was found that the existence of a proper class of Woodin cardinals was equiconsistent to AD. This is a significant weakening of superstrongness.

Definitions

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.

Elementary Embedding 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. [1]

Extender Definition

A cardinal $\kappa$ is $n$-superstrong (or $n$-fold superstrong when referring to the $n$-fold variants) 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 $Ult_{\mathcal{E}}(V)$). [1]

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

Relation to other large cardinal notions

  • The consistency strength of $n$-superstrongness follows the double helix pattern. [2] Specifically:
  • The consistency strength of the existence of a superstrong cardinal is less than that of the existence of a $\kappa$ which is $2^\kappa$-supercompact. [1]
  • If $\kappa$ is superstrong, then the necessary transitive class $M$ has $M\cap V_{j(\kappa)}$ satisfying $\kappa$'s strongness (although $\kappa$ may not be strong in $V$).
  • There are many measurable cardinals above a superstrong cardinal. [1]
  • Every $n$-huge cardinal is $n$-superstrong. [1]
  • If $\kappa$ is strong but there is a superstrong cardinal at least the size of $\kappa$, then there are also $\kappa$ superstrong cardinals less than $\kappa$. [1]
  • Every $1$-extendible cardinal is superstrong and has a normal measure containing all of the superstrongs less than said $1$-extendible. [1]
  • Every $\kappa$ which is $2^\kappa$-supercompact is larger than a superstrong cardinal and has a normal measure containing all of the superstrongs less than it. [1]
  • Every superstrong cardinal is Woodin and has a normal measure containing all of the Woodin cardinals less than it. [1]
  • Superstrongness is consistency-wise stronger than Hyper-Woodinness. [1]
  • Superstrongness is not Laver indestructible. [3]

References

  1. 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
  2. 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
  3. 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
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