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. | + | 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>. |
− | + | 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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+ | ==Ineffable cardinals and the constructible universe== | ||
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+ | Ineffable cardinals are downward absolute to $L$. <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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+ | ==Relations with other large 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. | ||
==Weakly ineffable cardinal== | ==Weakly ineffable cardinal== |
Revision as of 13:48, 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 $|\alpha|$.
Contents
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 limits of weakly compact cardinals.
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
- Jensen, Ronald and Kunen, Kenneth. Some combinatorial properties of $L$ and $V$. Unpublished, 1969. www bibtex