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>. | + | 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 | + | 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 $L$. | + | 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> | + | |
+ | 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 | + | * 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 $\kappa$. | + | 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 $\kappa$. |
+ | |||
+ | * Weakly ineffable cardinals are downward absolute to $L$. | ||
+ | * | ||
==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 $|\alpha|$.
Contents
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
- Measurable cardinals are ineffable and stationary limits of ineffable cardinals.
- Ineffable cardinals are $\Pi^1_2$-indescribable [1] and hence limits of weakly compact cardinals [3].
- The least ineffable cardinal is larger than the least totally indescribable cardinal. [1]
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
- Jensen, Ronald and Kunen, Kenneth. Some combinatorial properties of $L$ and $V$. Unpublished, 1969. www bibtex
- Jech, Thomas J. Set Theory. Third, Springer-Verlag, Berlin, 2003. (The third millennium edition, revised and expanded) www bibtex