Difference between revisions of "Ineffable"

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 $\kappa$.