Difference between revisions of "Strong"

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{{DISPLAYTITLE: Strong cardinal}}
 
{{DISPLAYTITLE: Strong cardinal}}
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[[Category:Large cardinal axioms]]
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[[Category:Critical points]]
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Strong cardinals were created as a weakening of [[supercompact]] cardinals introduced by Dodd and Jensen in 1982 <cite>Jech2003:SetTheory</cite>. They are defined as a strengthening of [[measurable|measurability]], being that they are critical points of [[elementary embedding|elementary embeddings]] $j:V\rightarrow M$ for some transitive inner model of [[ZFC]] $M$. Hypermeasurability is a weakening of strongness (the property of being a strong cardinal is often called strongness), although if $\lambda=\beth_\lambda$ then a cardinal is $\lambda$-strong iff it is $\lambda$-hypermeasurable.
  
A cardinal $\kappa$ is ''strong'' if it is $\theta$-strong for every ordinal $\theta$, meaning that there is an elementary embedding $j:V\to M$ of the universe $V$ into a transitive proper class $M$, having critical point $\kappa$ and with $V_\theta\subset M$. One may without loss also suppose that $j(\kappa)\gt\theta$.
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== Definitions of Strongness ==
  
Every strong cardinal is $\Sigma_2$-[[reflecting]], because if any $V_\theta$ satisfies some sentence $\sigma$, then if $j:V\to M$ witnesses $\theta$-strongness, then $M$ satisfies that some ordinal $\theta$ below $j(\kappa)$ has $V_\theta\models\sigma$, and so by elementarity $V$ satisifies that some $\theta\lt\kappa$ has $V_\theta\models\sigma$, as desired.  
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There are multiple equivalent definitions of strongness, using [[elementary embedding|elementary embeddings]] and [[extender|extenders]].
  
== Hypermeasurability ==
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=== Elementary Embedding Characterization ===
  
To give a more refined hiearachy, a cardinal $\kappa$ is ''$A$-strong'' or equivalently ''$A$-hypermeasurable'', if there is an elementary embedding $j:V\to M$ of the universe $V$ into a transtive class $M$, with critical point $\kappa$, for which $A\in M$.  
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A cardinal $\kappa$ is '''$\gamma$-strong''' iff it is the critical point of some elementary embedding $j:V\rightarrow M$ for some transitive class $M$ such that $V_\gamma\subset M$. A cardinal $\kappa$ is '''strong''' iff it is $\gamma$-strong for each $\gamma$, iff it is $\gamma$-strong for arbitrarily large $\gamma$, iff for each set $x$, $\kappa$ is the critical point of some elementary embedding $j:V\rightarrow M$ for some transitive class $M$ such that $x\in M$.
  
When $\delta$ is a cardinal, we say that $\kappa$ is ''$\delta$-hypermeasurable'' to mean really that $\kappa$ is $H_\delta$-strong. (Note, this terminology is technically ambiguous, since one could also interpret $\delta$-hypermeasurability as asserting merely that $\delta\in M$, but this reading is not the intended reading, and the ambiguity is avoided by realizing that $M$ contains all ordinals, and so this latter notion is not needed.)
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More intuitively, there are many elementary embeddings from $V$ into a transitive class which have critical point $\kappa$.
  
Note that the concept of $\theta$-strong is indexed by ordinals $\theta$ and the concept of $\delta$-hypermeasurable is indexed by cardinals. The latter concept provides a more refined hierarchy, particularly when the GCH fails.
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=== Extender Characterization ===
  
* A cardinal $\kappa$ is $\theta$-strong if and only if $\kappa$ is $\beth_\theta$-hypermeasurable.
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A cardinal $\kappa$ is '''strong''' iff it is [[uncountable]] and for every $\lambda\geq\kappa$, there is a $(\kappa,\beth_\lambda^+)$-extender $E$ such that, letting the [[ultrapower]] of $V$ by $E$ be called $Ult_E$ and the canonical ultrapower embedding from $V$ to $Ult_E$ be called $j$, $V_\lambda\subseteq Ult_E$ and $\lambda<j(\kappa)$. <cite>Jech2003:SetTheory</cite>
* In particular, $\kappa$ is $\mathcal{P}^2(\kappa)$-hypermeasurable if and only if it is $\kappa+2$-strong. This hypothesis appears in many theorems.
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* A cardinal $\kappa$ is [[measurable]] if and only if it is $\kappa^+$-hypermeasurable, since $\mathcal{P}(\kappa)\subset M$ for any $j:V\to M$ with critical point $\kappa$.  
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{{stub}}
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Once again, a more intuitive way to think about strongness is that there are many $(\kappa,\lambda)$-extenders $E$.
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== Definitions of Hypermeasurability ==
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The definitions of hypermeasurability are very similar to the definitions of strongness, mainly because hypermeasurability is a generalized version of strongness. The intuition behind each definition is also very similar to that of the matching definitions of strongness.
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=== Elementary Embedding Characterization ===
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A cardinal $\kappa$ is '''$x$-hypermeasurable''' for a set $x$ iff it is the critical point of some elementary embedding $j:V\rightarrow M$ for some transitive class $M$ such that $x\in M$. A cardinal $\kappa$ is '''$\lambda$-hypermeasurable''' iff it is $H_\lambda$-hypermeasurable (where $H_\lambda$ is the set of all sets of [[Hereditary Cardinality|hereditary cardinality]] less than $\lambda$).
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Note that a cardinal is $\gamma$-strong iff it is $x$-hypermeasurable for every $x\in V_\gamma$ (iff it is $V_\gamma$-hypermeasurable as well) and a cardinal is strong iff it is $x$-hypermeasurable for every $x$.
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== Facts about Strongness and Hypermeasurability ==
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Here is a list of facts about these cardinals:
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*A cardinal $\kappa$ is $\gamma$-strong if and only if $\kappa$ is $\beth_\gamma$-hypermeasurable, by definition.
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*In particular, $\kappa$ is $\mathcal{P}^2(\kappa)$-hypermeasurable if and only if it is $\kappa+2$-strong. This hypothesis appears in many theorems.
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*A cardinal $\kappa$ is [[measurable]] if and only if it is $\kappa^+$-hypermeasurable, since $\mathcal{P}(\kappa)\subset M$ for any $j:V\to M$ with critical point $\kappa$.
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*If there is an $x$-hypermeasurable cardinal, then $V\neq L[x]$. <cite>Jech2003:SetTheory</cite> 
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*Every [[Woodin]] cardinal $\kappa$ is strong and has $\kappa$ strong cardinals below it, as well as being a stationary limit of $\{\lambda<\kappa:\lambda\text{ is }<\kappa\text{ strong }\}$ (in this case, that means $\lambda$ is $\gamma$-strong for every $\gamma<\kappa$). The least Woodin cardinal is not [[weakly compact]], so it follows that not every strong cardinal is weakly compact. <cite>Jech2003:SetTheory</cite>
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*The [[Mitchell rank]] of any strong cardinal $o(\kappa)=(2^\kappa)^+$. <cite>Jech2003:SetTheory</cite>
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*Any strong cardinal is [[reflecting|$\Sigma_2$-reflecting]]. <cite>Jech2003:SetTheory</cite>
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*Assuming both a strong cardinal and a [[superstrong]] cardinal exist, then the least strong cardinal has $\kappa$ superstrong cardinals below it.
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== Core Model up to Strongness ==
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Dodd and Jensen created a [[core model]] based on sequences of [[extender|extenders]] of strong cardinals. They constructed a sequence of extenders $\mathcal{E}$ such that: <cite>Jech2003:SetTheory</cite>
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*[[L|$L[\mathcal{E}]$]] is an inner model of [[ZFC]].
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*$L[\mathcal{E}]$ satisfies [[GCH]], the square principle, and the existence of a $\Sigma_3^1$ well-ordering of $\mathbb{R}$.
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*$L[\mathcal{E}]$ satisfies that $\mathcal{E}$ witnesses the existence of a strong cardinal (looking at $\mathcal{E}$ as a function rather than a sequence).
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*If there does not exist an inner model of the existence of a strong cardinal, then:
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**There is no nontrivial elementary embedding $j:L[\mathcal{E}]\rightarrow L[\mathcal{E}]$
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**If $\kappa$ is a singular [[Beth|strong limit]] cardinal then $(\kappa^+)^{L[\mathcal{E}]}=\kappa^+$
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As one can see, $L[\mathcal{E}]$ is a core model up to strongness. Dodd and Jensen also constructed a "sharp" defined analogously to [[Zero sharp|$0^{\#}$]], but instead of using $L$ one uses $L[\mathcal{E}]$. They then showed that there is a nontrivial elementary embedding $j:L[\mathcal{E}]\rightarrow L[\mathcal{E}]$ iff such a real exists. <cite>Jech2003:SetTheory</cite> This real is commonly referred to as ''the sharp for a strong cardinal''.
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{{References}}

Revision as of 22:16, 6 November 2017

Strong cardinals were created as a weakening of supercompact cardinals introduced by Dodd and Jensen in 1982 [1]. They are defined as a strengthening of measurability, being that they are critical points of elementary embeddings $j:V\rightarrow M$ for some transitive inner model of ZFC $M$. Hypermeasurability is a weakening of strongness (the property of being a strong cardinal is often called strongness), although if $\lambda=\beth_\lambda$ then a cardinal is $\lambda$-strong iff it is $\lambda$-hypermeasurable.

Definitions of Strongness

There are multiple equivalent definitions of strongness, using elementary embeddings and extenders.

Elementary Embedding Characterization

A cardinal $\kappa$ is $\gamma$-strong iff it is the critical point of some elementary embedding $j:V\rightarrow M$ for some transitive class $M$ such that $V_\gamma\subset M$. A cardinal $\kappa$ is strong iff it is $\gamma$-strong for each $\gamma$, iff it is $\gamma$-strong for arbitrarily large $\gamma$, iff for each set $x$, $\kappa$ is the critical point of some elementary embedding $j:V\rightarrow M$ for some transitive class $M$ such that $x\in M$.

More intuitively, there are many elementary embeddings from $V$ into a transitive class which have critical point $\kappa$.

Extender Characterization

A cardinal $\kappa$ is strong iff it is uncountable and for every $\lambda\geq\kappa$, there is a $(\kappa,\beth_\lambda^+)$-extender $E$ such that, letting the ultrapower of $V$ by $E$ be called $Ult_E$ and the canonical ultrapower embedding from $V$ to $Ult_E$ be called $j$, $V_\lambda\subseteq Ult_E$ and $\lambda<j(\kappa)$. [1]

Once again, a more intuitive way to think about strongness is that there are many $(\kappa,\lambda)$-extenders $E$.

Definitions of Hypermeasurability

The definitions of hypermeasurability are very similar to the definitions of strongness, mainly because hypermeasurability is a generalized version of strongness. The intuition behind each definition is also very similar to that of the matching definitions of strongness.

Elementary Embedding Characterization

A cardinal $\kappa$ is $x$-hypermeasurable for a set $x$ iff it is the critical point of some elementary embedding $j:V\rightarrow M$ for some transitive class $M$ such that $x\in M$. A cardinal $\kappa$ is $\lambda$-hypermeasurable iff it is $H_\lambda$-hypermeasurable (where $H_\lambda$ is the set of all sets of hereditary cardinality less than $\lambda$).

Note that a cardinal is $\gamma$-strong iff it is $x$-hypermeasurable for every $x\in V_\gamma$ (iff it is $V_\gamma$-hypermeasurable as well) and a cardinal is strong iff it is $x$-hypermeasurable for every $x$.

Facts about Strongness and Hypermeasurability

Here is a list of facts about these cardinals:

  • A cardinal $\kappa$ is $\gamma$-strong if and only if $\kappa$ is $\beth_\gamma$-hypermeasurable, by definition.
  • In particular, $\kappa$ is $\mathcal{P}^2(\kappa)$-hypermeasurable if and only if it is $\kappa+2$-strong. This hypothesis appears in many theorems.
  • A cardinal $\kappa$ is measurable if and only if it is $\kappa^+$-hypermeasurable, since $\mathcal{P}(\kappa)\subset M$ for any $j:V\to M$ with critical point $\kappa$.
  • If there is an $x$-hypermeasurable cardinal, then $V\neq L[x]$. [1]
  • Every Woodin cardinal $\kappa$ is strong and has $\kappa$ strong cardinals below it, as well as being a stationary limit of $\{\lambda<\kappa:\lambda\text{ is }<\kappa\text{ strong }\}$ (in this case, that means $\lambda$ is $\gamma$-strong for every $\gamma<\kappa$). The least Woodin cardinal is not weakly compact, so it follows that not every strong cardinal is weakly compact. [1]
  • The Mitchell rank of any strong cardinal $o(\kappa)=(2^\kappa)^+$. [1]
  • Any strong cardinal is $\Sigma_2$-reflecting. [1]
  • Assuming both a strong cardinal and a superstrong cardinal exist, then the least strong cardinal has $\kappa$ superstrong cardinals below it.

Core Model up to Strongness

Dodd and Jensen created a core model based on sequences of extenders of strong cardinals. They constructed a sequence of extenders $\mathcal{E}$ such that: [1]

  • $L[\mathcal{E}]$ is an inner model of ZFC.
  • $L[\mathcal{E}]$ satisfies GCH, the square principle, and the existence of a $\Sigma_3^1$ well-ordering of $\mathbb{R}$.
  • $L[\mathcal{E}]$ satisfies that $\mathcal{E}$ witnesses the existence of a strong cardinal (looking at $\mathcal{E}$ as a function rather than a sequence).
  • If there does not exist an inner model of the existence of a strong cardinal, then:
    • There is no nontrivial elementary embedding $j:L[\mathcal{E}]\rightarrow L[\mathcal{E}]$
    • If $\kappa$ is a singular strong limit cardinal then $(\kappa^+)^{L[\mathcal{E}]}=\kappa^+$

As one can see, $L[\mathcal{E}]$ is a core model up to strongness. Dodd and Jensen also constructed a "sharp" defined analogously to $0^{\#}$, but instead of using $L$ one uses $L[\mathcal{E}]$. They then showed that there is a nontrivial elementary embedding $j:L[\mathcal{E}]\rightarrow L[\mathcal{E}]$ iff such a real exists. [1] This real is commonly referred to as the sharp for a strong cardinal.


References

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