Difference between revisions of "Ineffable"

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==Relations with other large cardinals==
 
==Relations with other large cardinals==
  
* [[Measurable]] cardinals are ineffable and stationary limits of ineffable cardinals.
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* [[Measurable]] cardinals are ineffable and stationary limits of ineffable cardinals.  
* Ineffable cardinals are $\Pi^1_2$-indescribable limits of weakly compact cardinals.
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* Ineffable cardinals are $\Pi^1_2$-indescribable and hence limits of weakly compact cardinals. <cite>JensenKunen1969:Ineffable</cite>
 
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* The least ineffable cardinal is larger than the least totally indescribable cardinal. <cite>JensenKunen1969:Ineffable</cite>
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==Weakly ineffable cardinal==
 
==Weakly ineffable cardinal==
  
 
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 $\kappa$.
 
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 $\kappa$.
 
{{References}}
 
{{References}}

Revision as of 14:05, 12 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].

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 $L$. [1] In $L$, an inaccessible cardinal $\kappa$ is ineffable if and only if there are no slim $\kappa$-Kurepa trees. [1]

Relations with other large cardinals

  • Measurable cardinals are ineffable and stationary limits of ineffable cardinals.
  • Ineffable cardinals are $\Pi^1_2$-indescribable and hence limits of weakly compact cardinals. [1]
  • The least ineffable cardinal is larger than the least totally indescribable cardinal. [1]

Weakly ineffable cardinal

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 $\kappa$.

References

  1. Jensen, Ronald and Kunen, Kenneth. Some combinatorial properties of $L$ and $V$. Unpublished, 1969. www   bibtex
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