Difference between revisions of "Superstrong"

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(Relation to other large cardinal notions)
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[[Category:Large cardinal axioms]]
 
[[Category:Large cardinal axioms]]
 
[[Category:Critical points]]
 
[[Category:Critical points]]
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.
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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 (ZF+)AD. After this, it was found that the existence of a proper class of [[Woodin]] cardinals was equiconsistent to AD. Woodin-ness is a significant weakening of superstrongness.
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''Most results in this article can be found in <cite>Kanamori2009:HigherInfinite</cite> unless indicated otherwise.''
  
 
== Definitions ==
 
== Definitions ==
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**super $n$-huge
 
**super $n$-huge
 
**$(n+1)$-superstrong
 
**$(n+1)$-superstrong
*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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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 $V$).
*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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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.
*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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Because of [[strong|strongness]]'s downward reflection properties, 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 $\kappa$. In particular the least superstrong is smaller than the least strong.
*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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Every [[extendible|$1$-extendible]] cardinal is superstrong and has a [[filter|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 [[supercompact|$2^\kappa$-supercompact]] is larger than the least superstrong cardinal and has a normal measure containing all of the superstrongs less than it.
*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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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 [[Woodin|Hyper-Woodinness]].
*If there is a superstrong cardinal, then in [[Constructible universe|$L[\mathbb{R}]$]], the [[axiom of determinacy]] holds. Thus: <cite>Jech2003:SetTheory</cite>
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**Every set of reals is Lebesgue measurable.
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If there is a superstrong cardinal, then in [[Constructible universe|$L[\mathbb{R}]$]], the [[axiom of determinacy]] holds. <cite>Jech2003:SetTheory</cite> Thus:
**Every set of reals has the Baire property.
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* Every set of reals is Lebesgue measurable.
**Every set of reals is a perfect set.
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* Every set of reals has the Baire property.
**There is no well-ordering of the reals.
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* Every set of reals is a perfect set.
*Although it may seem like superstrong cardinals are quite large, letting $\kappa$ be superstrong, $\kappa$ can be [[forcing|forced]] to $\aleph_2$ with an $\omega$-distributive, $\kappa$-c.c. notion of forcing, and in fact, in this new universe there is a normal $\aleph_2$-saturated ideal on $\omega_1$. More intuitively, one can easily create force universes such that $\kappa$ is no longer superstrong and is in fact quite small. <cite>Jech2003:SetTheory</cite>
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* There is no well-ordering of the reals.
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* Although it may seem like superstrong cardinals are quite large, letting $\kappa$ be superstrong, $\kappa$ can be [[forcing|forced]] to $\aleph_2$ with an $\omega$-distributive, $\kappa$-c.c. notion of forcing, and in fact, in this new universe there is a normal $\aleph_2$-saturated ideal on $\omega_1$. More intuitively, one can easily create force universes such that $\kappa$ is no longer superstrong and is in fact quite small. <cite>Jech2003:SetTheory</cite>
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Superstrongness is not Laver indestructible. <cite>BagariaHamkinsTsaprounisUsuba:SuperstrongAndOtherLargeCardinalsAreNeverLaverIndestructible</cite>  
  
 
{{References}}
 
{{References}}

Revision as of 11:23, 9 November 2017

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 (ZF+)AD. After this, it was found that the existence of a proper class of Woodin cardinals was equiconsistent to AD. Woodin-ness is a significant weakening of superstrongness.

Most results in this article can be found in [1] unless indicated otherwise.

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

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 $V$).

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.

Because of strongness's downward reflection properties, 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 $\kappa$. In particular the least superstrong is smaller than the least strong.

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. [3] Thus:

  • Every set of reals is Lebesgue measurable.
  • Every set of reals has the Baire property.
  • Every set of reals is a perfect set.
  • There is no well-ordering of the reals.
  • Although it may seem like superstrong cardinals are quite large, letting $\kappa$ be superstrong, $\kappa$ can be forced to $\aleph_2$ with an $\omega$-distributive, $\kappa$-c.c. notion of forcing, and in fact, in this new universe there is a normal $\aleph_2$-saturated ideal on $\omega_1$. More intuitively, one can easily create force universes such that $\kappa$ is no longer superstrong and is in fact quite small. [3]

Superstrongness is not Laver indestructible. [4]

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. Jech, Thomas J. Set Theory. Third, Springer-Verlag, Berlin, 2003. (The third millennium edition, revised and expanded) www   bibtex
  4. 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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