Difference between revisions of "Ineffable"
Line 15: | Line 15: | ||
* [[Measurable]] cardinals are ineffable and stationary limits of ineffable cardinals. | * [[Measurable]] cardinals are ineffable and stationary limits of ineffable cardinals. | ||
− | * Ineffable cardinals are | + | * Ineffable cardinals are $\Pi^1_2$-[[indescribable]] <cite>JensenKunen1969:Ineffable</cite>. |
* Ineffable cardinals are limits of [[totally indescribable]] cardinals. <cite>JensenKunen1969:Ineffable</cite> (<cite>AbramsonHarringtonKleinbergZwicker1977:FlippingProperties</cite> for proof) | * Ineffable cardinals are limits of [[totally indescribable]] cardinals. <cite>JensenKunen1969:Ineffable</cite> (<cite>AbramsonHarringtonKleinbergZwicker1977:FlippingProperties</cite> for proof) | ||
Line 23: | Line 23: | ||
* Weakly ineffable cardinals are downward absolute to $L$. <cite>JensenKunen1969:Ineffable</cite> | * Weakly ineffable cardinals are downward absolute to $L$. <cite>JensenKunen1969:Ineffable</cite> | ||
− | * Weakly ineffable cardinals are | + | * Weakly ineffable cardinals are $\Pi_1^1$-[[indescribable]]. <cite>JensenKunen1969:Ineffable</cite> |
* Ineffable cardinals are limits of weakly ineffable cardinals. | * Ineffable cardinals are limits of weakly ineffable cardinals. | ||
* Weakly ineffable cardinals are limits of [[totally indescribable]] cardinals. <cite>JensenKunen1969:Ineffable</cite> (<cite>AbramsonHarringtonKleinbergZwicker1977:FlippingProperties</cite> for proof) | * Weakly ineffable cardinals are limits of [[totally indescribable]] cardinals. <cite>JensenKunen1969:Ineffable</cite> (<cite>AbramsonHarringtonKleinbergZwicker1977:FlippingProperties</cite> for proof) | ||
Line 35: | Line 35: | ||
* Subtle cardinals are limits of [[totally indescribable]] cardinals. <cite>JensenKunen1969:Ineffable</cite> | * Subtle cardinals are limits of [[totally indescribable]] cardinals. <cite>JensenKunen1969:Ineffable</cite> | ||
* The least subtle cardinal is not weakly compact as it is $\Pi_1^1$-describable. | * The least subtle cardinal is not weakly compact as it is $\Pi_1^1$-describable. | ||
+ | |||
+ | ==Completely ineffable cardinal== | ||
+ | |||
+ | Completely ineffable cardinals were introduced in <cite>AbramsonHarringtonKleinbergZwicker1977:FlippingProperties</cite> as a strenthening of ineffable cardinals. Define that a collection $R\subseteq P(\kappa)$ is a stationary | ||
+ | class if | ||
+ | * $R\neq\emptyset$, | ||
+ | * for all $A\in R$, $A$ is stationary in $\kappa$, | ||
+ | * if $A\in R$ and $B\supseteq A$, then $B\in R$. | ||
+ | A cardinal $\kappa$ is completely ineffable if there is a stationary class $R$ such that for every $A\in R$ and $F:[A]^2\to | ||
+ | 2$, there is $H\in R$ such that $F\upharpoonright [H]^2$ is constant. | ||
+ | |||
+ | * Completely ineffable cardinals are downward absolute to $L$. <cite>AbramsonHarringtonKleinbergZwicker1977:FlippingProperties</cite> | ||
+ | * Completely ineffable cardinals are limits of ineffable cardinals. <cite>AbramsonHarringtonKleinbergZwicker1977:FlippingProperties</cite> | ||
{{References}} | {{References}} |
Revision as of 12:50, 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].
- Ineffable cardinals are limits of totally indescribable cardinals. [1] ([3] for proof)
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 $\kappa$. If $\kappa$ is weakly ineffable, then $\diamondsuit_\kappa$ holds.
- Weakly ineffable cardinals are downward absolute to $L$. [1]
- Weakly ineffable cardinals are $\Pi_1^1$-indescribable. [1]
- Ineffable cardinals are limits of weakly ineffable cardinals.
- Weakly ineffable cardinals are limits of totally indescribable cardinals. [1] ([3] for proof)
Subtle cardinal
Subtle cardinals were introduced by Jensen and Kunen in [1] as a weakening of weakly ineffable cardinals. A uncountable regular cardinal $\kappa$ is subtle if for every for every $\langle A_\alpha\mid \alpha<\kappa\rangle$ with $A_\alpha\subseteq \alpha$ and every closed unbounded $C\subseteq\kappa$ there are $\alpha<\beta$ in $C$ such that $A_\beta\cap\alpha=A_\alpha$. If $\kappa$ is subtle, then $\diamondsuit_\kappa$ holds.
- Subtle cardinals are downward absolute to $L$. [1]
- Weakly ineffable cardinals are limits of subtle cardinals. [1]
- Subtle cardinals are limits of totally indescribable cardinals. [1]
- The least subtle cardinal is not weakly compact as it is $\Pi_1^1$-describable.
Completely ineffable cardinal
Completely ineffable cardinals were introduced in [3] as a strenthening of ineffable cardinals. Define that a collection $R\subseteq P(\kappa)$ is a stationary class if
- $R\neq\emptyset$,
- for all $A\in R$, $A$ is stationary in $\kappa$,
- if $A\in R$ and $B\supseteq A$, then $B\in R$.
A cardinal $\kappa$ is completely ineffable if there is a stationary class $R$ such that for every $A\in R$ and $F:[A]^2\to 2$, there is $H\in R$ such that $F\upharpoonright [H]^2$ is constant.
- Completely ineffable cardinals are downward absolute to $L$. [3]
- Completely ineffable cardinals are limits of ineffable cardinals. [3]
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
- Abramson, Fred and Harrington, Leo and Kleinberg, Eugene and Zwicker, William. Flipping properties: a unifying thread in the theory of large cardinals. Ann Math Logic 12(1):25--58, 1977. MR bibtex