Difference between revisions of "Ineffable"

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{{DISPLAYTITLE: Ineffable cardinal}}
 
{{DISPLAYTITLE: Ineffable cardinal}}
  
Ineffable cardinals were introduced by Jensen and Kunen in <cite>JensenKunen1969:Ineffable</cite>. An uncountable regular cardinal $\kappa$ is ineffable if for every sequence $\langle A_\alpha\mid \alpha<\kappa\rangle$ with $A_\alpha\subseteq \alpha$ there is $A\subseteq\kappa$ such that the set $S=\{\alpha<\kappa\mid A\cap \kappa=A_\alpha\}$ is stationary. Equivalently an uncountable regular $\kappa$ is ineffable if and only if for every function $F:[\kappa]^2\rightarrow 2$ there is a stationary $H\subseteq\kappa$ such that $F\upharpoonright [H]^2$ is constant <cite>JensenKunen1969:Ineffable</cite>.
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Ineffable cardinals were introduced by Jensen and Kunen in <cite>JensenKunen1969:Ineffable</cite>. An uncountable regular cardinal $\kappa$ is ineffable if for every sequence $\langle A_\alpha\mid \alpha<\kappa\rangle$ with $A_\alpha\subseteq \alpha$ there is $A\subseteq\kappa$ such that the set $S=\{\alpha<\kappa\mid A\cap \kappa=A_\alpha\}$ is stationary. Equivalently an uncountable regular $\kappa$ is ineffable if and only if for every function $F:[\kappa]^2\rightarrow 2$ there is a stationary $H\subseteq\kappa$ such that $F\upharpoonright [H]^2$ is constant <cite>JensenKunen1969:Ineffable</cite>. This second characterization strengthens a characterization of weakly compact cardinals which requires that there exist such an $H$ of size $\kappa$.
  
If $\kappa$ is ineffable, then $\diamondsuit_\kappa$ holds and there cannot be a slim $\kappa$-Kurepa tree. <cite>JensenKunen1969:Ineffable</cite> A $\kappa$-Kurepa tree is a tree of height $\kappa$ having levels of size less than $\kappa$ and at least $\kappa^+$-many branches. A $\kappa$-Kurepa tree is slim if every infinite level $\alpha$ has size at most $|\alpha|$.  
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If $\kappa$ is ineffable, then $\diamondsuit_\kappa$ holds and there cannot be a slim $\kappa$-Kurepa tree <cite>JensenKunen1969:Ineffable</cite> . A $\kappa$-Kurepa tree is a tree of height $\kappa$ having levels of size less than $\kappa$ and at least $\kappa^+$-many branches. A $\kappa$-Kurepa tree is slim if every infinite level $\alpha$ has size at most $|\alpha|$.  
  
  
 
==Ineffable cardinals and the constructible universe==
 
==Ineffable cardinals and the constructible universe==
  
Ineffable cardinals are downward absolute to $L$. <cite>JensenKunen1969:Ineffable</cite>
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Ineffable cardinals are downward absolute to $L$. In $L$, an inaccessible cardinal $\kappa$ is ineffable if and only if there are no slim $\kappa$-Kurepa trees. Thus, for inaccessible cardinals, in $L$, ineffability is completely characterized using slim Kurepa trees. <cite>JensenKunen1969:Ineffable</cite>
In $L$, an inaccessible cardinal $\kappa$ is ineffable if and only if there are no slim $\kappa$-Kurepa trees. <cite>JensenKunen1969:Ineffable</cite>
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If [[zero sharp | $0^\sharp$]] exists, then every Silver indiscernible is ineffable in $L$. <cite>Jech2003:SetTheory</cite>
  
 
==Relations with other large cardinals==
 
==Relations with other large cardinals==
  
 
* [[Measurable]] cardinals are ineffable and stationary limits of ineffable cardinals.  
 
* [[Measurable]] cardinals are ineffable and stationary limits of ineffable cardinals.  
* Ineffable cardinals are [[indescribable|$\Pi^1_2$-indescribable]] and hence limits of weakly compact cardinals. <cite>JensenKunen1969:Ineffable</cite>
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* Ineffable cardinals are [[indescribable|$\Pi^1_2$-indescribable]] <cite>JensenKunen1969:Ineffable</cite> and hence limits of weakly compact cardinals <cite>Devlin1984:Constructibility</cite>.
 
* The least ineffable cardinal is larger than the least [[totally indescribable]] cardinal. <cite>JensenKunen1969:Ineffable</cite>
 
* The least ineffable cardinal is larger than the least [[totally indescribable]] cardinal. <cite>JensenKunen1969:Ineffable</cite>
 
   
 
   
 
==Weakly ineffable cardinal==
 
==Weakly ineffable cardinal==
  
Weakly ineffable cardinals (also called almost ineffable) were introduced by Jensen and Kunen in <cite>JensenKunen1969:Ineffable</cite>. An uncountable regular cardinal $\kappa$ is weakly ineffable if for every sequence $\langle A_\alpha\mid \alpha<\kappa\rangle$ with $A_\alpha\subseteq \alpha$ there is $A\subseteq\kappa$ such that the set $S=\{\alpha<\kappa\mid A\cap \kappa=A_\alpha\}$ has size $\kappa$.
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Weakly ineffable cardinals (also called almost ineffable) were introduced by Jensen and Kunen in <cite>JensenKunen1969:Ineffable</cite> as a weakening of ineffable cardinals. An uncountable regular cardinal $\kappa$ is weakly ineffable if for every sequence $\langle A_\alpha\mid \alpha<\kappa\rangle$ with $A_\alpha\subseteq \alpha$ there is $A\subseteq\kappa$ such that the set $S=\{\alpha<\kappa\mid A\cap \kappa=A_\alpha\}$ has size $\kappa$.
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* Weakly ineffable cardinals are downward absolute to $L$.
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*
  
 
==Subtle cardinal==
 
==Subtle cardinal==
  
 
{{References}}
 
{{References}}

Revision as of 09:34, 13 January 2012


Ineffable cardinals were introduced by Jensen and Kunen in [1]. An uncountable regular cardinal $\kappa$ is ineffable if for every sequence $\langle A_\alpha\mid \alpha<\kappa\rangle$ with $A_\alpha\subseteq \alpha$ there is $A\subseteq\kappa$ such that the set $S=\{\alpha<\kappa\mid A\cap \kappa=A_\alpha\}$ is stationary. Equivalently an uncountable regular $\kappa$ is ineffable if and only if for every function $F:[\kappa]^2\rightarrow 2$ there is a stationary $H\subseteq\kappa$ such that $F\upharpoonright [H]^2$ is constant [1]. This second characterization strengthens a characterization of weakly compact cardinals which requires that there exist such an $H$ of size $\kappa$.

If $\kappa$ is ineffable, then $\diamondsuit_\kappa$ holds and there cannot be a slim $\kappa$-Kurepa tree [1] . A $\kappa$-Kurepa tree is a tree of height $\kappa$ having levels of size less than $\kappa$ and at least $\kappa^+$-many branches. A $\kappa$-Kurepa tree is slim if every infinite level $\alpha$ has size at most $|\alpha|$.


Ineffable cardinals and the constructible universe

Ineffable cardinals are downward absolute to $L$. In $L$, an inaccessible cardinal $\kappa$ is ineffable if and only if there are no slim $\kappa$-Kurepa trees. Thus, for inaccessible cardinals, in $L$, ineffability is completely characterized using slim Kurepa trees. [1]

If $0^\sharp$ exists, then every Silver indiscernible is ineffable in $L$. [2]

Relations with other large cardinals

Weakly ineffable cardinal

Weakly ineffable cardinals (also called almost ineffable) were introduced by Jensen and Kunen in [1] as a weakening of ineffable cardinals. An uncountable regular cardinal $\kappa$ is weakly ineffable if for every sequence $\langle A_\alpha\mid \alpha<\kappa\rangle$ with $A_\alpha\subseteq \alpha$ there is $A\subseteq\kappa$ such that the set $S=\{\alpha<\kappa\mid A\cap \kappa=A_\alpha\}$ has size $\kappa$.

  • Weakly ineffable cardinals are downward absolute to $L$.

Subtle cardinal

References

  1. Jensen, Ronald and Kunen, Kenneth. Some combinatorial properties of $L$ and $V$. Unpublished, 1969. www   bibtex
  2. Jech, Thomas J. Set Theory. Third, Springer-Verlag, Berlin, 2003. (The third millennium edition, revised and expanded) www   bibtex
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