Difference between revisions of "Tall"

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{{DISPLAYTITLE: Tall cardinal}}
 
{{DISPLAYTITLE: Tall cardinal}}
  
A cardinal $\kappa$ is ''tall'' if for every ordinal $\theta$ there is an [[elementary embedding]] $j:V\to M$ into a transitive class $M$ with critical point $\kappa$ such that $j(\kappa)>\theta$ and $M^\kappa\subset M$. Every [[strong]] cardinal is tall and every [[strongly compact]] cardinal is tall, but [[measurable]] cardinals are not necessarily tall. It is relatively consistent, however, that the least measurable cardinal is tall. Nevertheless, the existence of a tall cardinal is equiconsistent with the existence of a [[strong]] cardinal. Any tall cardinal $\kappa$ can be made indestructible by a variety of forcing notions,
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== Tall Cardinals ==
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A cardinal $\kappa$ is '''$\theta$-tall''' iff there is an [[elementary embedding]] $j:V\to M$ into a transitive class $M$ with critical point $\kappa$ such that $j(\kappa)>\theta$ and $M^\kappa\subset M$. $\kappa$ is '''tall''' iff it is $\theta$-tall for every $\theta$; i.e. $j(\kappa)$ can be made arbitrarily large. Every [[strong]] cardinal is tall and every [[strongly compact]] cardinal is tall, but [[measurable]] cardinals are not necessarily tall. It is relatively consistent, however, that the least measurable cardinal is tall. Nevertheless, the existence of a tall cardinal is equiconsistent with the existence of a [[strong]] cardinal. Any tall cardinal $\kappa$ can be made indestructible by a variety of forcing notions,
 
including forcing that pumps up the value of $2^\kappa$ as high as desired. See <cite>Hamkins2009:TallCardinals</cite>.
 
including forcing that pumps up the value of $2^\kappa$ as high as desired. See <cite>Hamkins2009:TallCardinals</cite>.
  
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== Strongly Tall Cardinals ==
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A cardinal $\kappa$ is '''strongly $\theta$-tall''' iff there is some [[filter|measure]] $U$ on a set $S$ witnessing $\kappa$'s $\theta$-tallness in the ultrapower of $V$ by $U$. More precisely, the ultrapower embedding $j:V\prec M$ has critical point $\kappa$, $M^\kappa\subset M$, and $j(\kappa)>\theta$. $\kappa$ is '''strongly tall''' iff it is strongly $\theta$-tall for every $\theta$. It is not known whether or not all [[strong]] cardinals are strongly tall, although every [[strongly compact|strongly $\theta$-compact]] cardinal is strongly $\theta$-tall. It is conjectured that strongly tall cardinals are equiconsistent with strong cardinals (and therefore with tall cardinals).
  
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=== Ultrapower Characterization ===
  
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$\kappa$ is strongly $\theta$-tall iff $\kappa$ is uncountable and there is some set $S$ and an [[filter|ultrafilter]] $U$ on $S$ with completeness $\kappa^+$ ($U$ is $\kappa$-complete but not $\kappa^+$-complete) such that, letting $j:V\prec M\cong Ult_U(V)$, $j(\kappa)>\theta$.
  
{{stub}}
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=== Ultrafilter Characterization ===
  
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$\kappa$ is strongly $\theta$-tall iff there is some set $S$, an [[filter|ultrafilter]] $U$ on $S$, and a class $H$ of functions $H_\alpha:S\rightarrow\text{Ord}$ for each ordinal $\alpha$ such that:
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#$\kappa$ is uncountable.
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#$U$ has completeness $\kappa^+$.
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#$H_0(x)=0$ for each $x\in S$.
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#For each $\alpha$ and each $f:S\rightarrow\text{Ord}$, $\{x\in S:f(x)<H_\alpha(x)\}\in U$ iff there is some $\beta<\alpha$ such that $\{x\in S:f(x)=\beta\}\in U$. That is, $f(x)<H_\alpha(x)$ almost everywhere iff there is some $\beta<\alpha$ such that $f(x)=\beta$ almost everywhere.
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#$\{x\in S:H_\theta(x)<\kappa\}\in U$. That is, $H_\theta(x)<\kappa$ almost everywhere.
  
 
{{References}}
 
{{References}}

Revision as of 23:33, 6 October 2018


Tall Cardinals

A cardinal $\kappa$ is $\theta$-tall iff there is an elementary embedding $j:V\to M$ into a transitive class $M$ with critical point $\kappa$ such that $j(\kappa)>\theta$ and $M^\kappa\subset M$. $\kappa$ is tall iff it is $\theta$-tall for every $\theta$; i.e. $j(\kappa)$ can be made arbitrarily large. Every strong cardinal is tall and every strongly compact cardinal is tall, but measurable cardinals are not necessarily tall. It is relatively consistent, however, that the least measurable cardinal is tall. Nevertheless, the existence of a tall cardinal is equiconsistent with the existence of a strong cardinal. Any tall cardinal $\kappa$ can be made indestructible by a variety of forcing notions, including forcing that pumps up the value of $2^\kappa$ as high as desired. See [1].

Strongly Tall Cardinals

A cardinal $\kappa$ is strongly $\theta$-tall iff there is some measure $U$ on a set $S$ witnessing $\kappa$'s $\theta$-tallness in the ultrapower of $V$ by $U$. More precisely, the ultrapower embedding $j:V\prec M$ has critical point $\kappa$, $M^\kappa\subset M$, and $j(\kappa)>\theta$. $\kappa$ is strongly tall iff it is strongly $\theta$-tall for every $\theta$. It is not known whether or not all strong cardinals are strongly tall, although every strongly $\theta$-compact cardinal is strongly $\theta$-tall. It is conjectured that strongly tall cardinals are equiconsistent with strong cardinals (and therefore with tall cardinals).

Ultrapower Characterization

$\kappa$ is strongly $\theta$-tall iff $\kappa$ is uncountable and there is some set $S$ and an ultrafilter $U$ on $S$ with completeness $\kappa^+$ ($U$ is $\kappa$-complete but not $\kappa^+$-complete) such that, letting $j:V\prec M\cong Ult_U(V)$, $j(\kappa)>\theta$.

Ultrafilter Characterization

$\kappa$ is strongly $\theta$-tall iff there is some set $S$, an ultrafilter $U$ on $S$, and a class $H$ of functions $H_\alpha:S\rightarrow\text{Ord}$ for each ordinal $\alpha$ such that:

  1. $\kappa$ is uncountable.
  2. $U$ has completeness $\kappa^+$.
  3. $H_0(x)=0$ for each $x\in S$.
  4. For each $\alpha$ and each $f:S\rightarrow\text{Ord}$, $\{x\in S:f(x)<H_\alpha(x)\}\in U$ iff there is some $\beta<\alpha$ such that $\{x\in S:f(x)=\beta\}\in U$. That is, $f(x)<H_\alpha(x)$ almost everywhere iff there is some $\beta<\alpha$ such that $f(x)=\beta$ almost everywhere.
  5. $\{x\in S:H_\theta(x)<\kappa\}\in U$. That is, $H_\theta(x)<\kappa$ almost everywhere.

References

  1. Hamkins, Joel David. Tall cardinals. MLQ Math Log Q 55(1):68--86, 2009. www   DOI   MR   bibtex
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