Ineffable cardinals were introduced by Jensen and Kunen in {% cite Jensen1969 %} and arose out of their study of $\diamondsuit$ principles. >[!info] Definition >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 $S=\{\alpha<\kappa\mid A\cap \alpha=A_\alpha\}$ 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 Jensen1969 %}. >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 {% cite Jensen1969 %}. 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 $\|\alpha\|$. An uncountable cardinal κ has the [normal filter property](Filter_property "Filter property") iff it is ineffable. {% cite Holy2018 %} ## 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. {% cite Jensen1969 %} If [$0^\sharp$](Zero%20sharp.md "Zero sharp") exists, then every Silver indiscernible is ineffable in $L$. {% cite Jech2003 %} [Ramsey](Ramsey.md "Ramsey") cardinals are stationary limits of completely ineffable cardinals, they are weakly ineffable, but the least Ramsey cardinal is not ineffable. Ineffable Ramsey cardinals are limits of Ramsey cardinals, because ineffable cardinals are $Π^1_2$-indescribable and being Ramsey is a $Π^1_2$-statement. The least strongly Ramsey cardinal also is not ineffable, but super weakly Ramsey cardinals are ineffable. $1$-iterable (=weakly Ramsey) cardinals are weakly ineffable and stationary limits of completely ineffable cardinals. The least $1$-iterable cardinal is not ineffable. {% cite Holy2018 Gitman2011 %} ## Relations with other large cardinals - [Measurable](Measurable.md "Measurable") cardinals are ineffable and stationary limits of ineffable cardinals. - [$\omega$-Erdős](Erdos.md "Erdos") cardinals are stationary limits of ineffable cardinals, but not ineffable since they are $\Pi_1^1$-describable. {% cite Jech2003 %} - Ineffable cardinals are $\Pi^1_2$-[indescribable](Indescribable.md "Indescribable") {% cite Jensen1969 %}. - Ineffable cardinals are limits of [totally indescribable](Totally_indescribable "Totally indescribable") cardinals. ({% cite Jensen1969 Abramson1977 %} for proof) - For a cardinal $κ=κ^{<κ}$, $κ$ is ineffable iff it is normal 0-[Ramsey](Ramsey.md "Ramsey"). {% cite Nielsen2018 %} ## Weakly ineffable cardinal Weakly ineffable cardinals (also called almost ineffable) were introduced by Jensen and Kunen in {% cite Jensen1969 %} 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 $S=\{\alpha<\kappa\mid A\cap \alpha=A_\alpha\}$ has size $\kappa$. If $\kappa$ is weakly ineffable, then $\diamondsuit_\kappa$ holds. - Weakly ineffable cardinals are downward absolute to $L$. {% cite Jensen1969 %} - Weakly ineffable cardinals are $\Pi_1^1$-[indescribable](Indescribable.md "Indescribable"). {% cite Jensen1969 %} - Ineffable cardinals are limits of weakly ineffable cardinals. - Weakly ineffable cardinals are limits of [totally indescribable](Totally_indescribable "Totally indescribable") cardinals. {% cite Jensen1969 %} ({% cite Abramson1977 %} for proof) - For a cardinal $κ=κ^{<κ}$, $κ$ is weakly ineffable iff it is genuine 0-[Ramsey](Ramsey.md "Ramsey"). {% cite Nielsen2018 %} ## Subtle cardinal Subtle cardinals were introduced by Jensen and Kunen in {% cite Jensen1969 %} 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 $L$. {% cite Jensen1969 %} - Weakly ineffable cardinals are limits of subtle cardinals. {% cite Jensen1969 %} - Subtle cardinals are stationary limits of [totally indescribable](Totally_indescribable "Totally indescribable") cardinals. {% cite Jensen1969 Friedman1998 %} - The least subtle cardinal is not weakly compact as it is $\Pi_1^1$-describable. - [$\alpha$-Erdős](Erdos.md "Erdos") cardinals are subtle. {% cite Jensen1969 %} - If $δ$ is a subtle cardinal, - the set of cardinals $κ$ below $δ$ that are [](Uplifting.md) in $V_δ$ is stationary. {% cite Hamkins2014a %} - for every class $\mathcal{A}$, in every club $B ⊆ δ$ there is $κ$ such that $\langle V_δ, \mathcal{A} ∩ V_δ \rangle \models \text{“$κ$ is $\mathcal{A}$-shrewd.”}$. {% cite Rathjen2006 %} (The set of cardinals $κ$ below $δ$ that are $\mathcal{A}$-[shrewd](Shrewd.md "Shrewd") in $V_δ$ is stationary.) - there is an $\eta$-shrewd cardinal below $δ$ for all $\eta < δ$. {% cite Rathjen2006 %} ## Ethereal cardinal *To be expanded.* ## $n$-ineffable cardinal The $n$-ineffable cardinals for $2\leq n<\omega$ were introduced by Baumgartner in {% cite Baumgartner1975 %} as a strengthening of ineffable cardinals. A cardinal is $n$-ineffable if for every function $F:\[\kappa\]^n\rightarrow 2$ there is a stationary $H\subseteq\kappa$ such that $F\upharpoonright \[H\]^n$ is constant. - $2$-ineffable cardinals are exactly the ineffable cardinals. - an $n+1$-ineffable cardinal is a stationary limit of $n$-ineffable cardinals. {% cite Baumgartner1975 %} A cardinal $\kappa$ is totally ineffable if it is $n$-ineffable for every $n$. - a $1$-iterable cardinal is a stationary limit of totally ineffable cardinals. (this follows from material in {% cite Gitman2011 %}) ### Helix (Information in this subsection come from {% cite Friedman1998 %} unless noted otherwise.) For $k \geq 1$ we define: - $\mathcal{P}(x)$ is the powerset (set of all subsets) of $x$. $\mathcal{P}_k(x)$ is the set of all subsets of $x$ with exactly $k$ elements. - $f:\mathcal{P}_k(\lambda) \to \mathcal{P}(\lambda)$ is regressive iff for all $A \in \mathcal{P}_k(\lambda)$, we have $f(A) \subseteq \min(A)$. - $E$ is $f$-homogenous iff $E \subseteq \lambda$ and for all $B,C \in \mathcal{P}_k(E)$, we have $f(B) \cap \min(B \cup C) = f(C) \cap \min(B \cup C)$. - $\lambda$ is $k$-subtle iff $\lambda$ is a limit ordinal and for all clubs $C \subseteq \lambda$ and regressive $f:\mathcal{P}_k(\lambda) \to \mathcal{P}(\lambda)$, there exists an $f$-homogenous $A \in \mathcal{P}_{k+1}(C)$. - $\lambda$ is $k$-almost ineffable iff $\lambda$ is a limit ordinal and for all regressive $f:\mathcal{P}_k(\lambda) \to \mathcal{P}(\lambda)$, there exists an $f$-homogenous $A \subseteq \lambda$ of cardinality $\lambda$. - $\lambda$ is $k$-ineffable iff $\lambda$ is a limit ordinal and for all regressive $f:\mathcal{P}_k(\lambda) \to \mathcal{P}(\lambda)$, there exists an $f$-homogenous stationary $A \subseteq \lambda$. $0$-subtle, $0$-almost ineffable and $0$-ineffable cardinals can be defined as “uncountable regular cardinals” because for $k \geq 1$ all three properties imply being uncountable regular cardinals. - For $k \geq 1$, if $\kappa$ is a $k$-ineffable cardinal, then $\kappa$ is $k$-almost ineffable and the set of $k$-almost ineffable cardinals is stationary in $\kappa$. - For $k \geq 1$, if $\kappa$ is a $k$-almost ineffable cardinal, then $\kappa$ is $k$-subtle and the set of $k$-subtle cardinals is stationary in $\kappa$. - For $k \geq 1$, if $\kappa$ is a $k$-subtle cardinal, then the set of $(k-1)$-ineffable cardinals is stationary in $\kappa$. - For $k \geq n \geq 0$, all $k$-ineffable cardinals are $n$-ineffable, all $k$-almost ineffable cardinals are $n$-almost ineffable and all $k$-subtle cardinals are $n$-subtle. This structure is similar to the [](N-fold_variants.md) and earlier known although smaller. {% cite Kentaro2007 %} ## Completely ineffable cardinal Completely ineffable cardinals were introduced in {% cite Abramson1977 %} as a strengthening 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\to2$, there is $H\in R$ such that $F\upharpoonright \[H\]^2$ is constant. Relations: - Completely ineffable cardinals are downward absolute to $L$. {% cite Abramson1977 %} - Completely ineffable cardinals are limits of ineffable cardinals. {% cite Abramson1977 %} - There are stationarily many completely ineffable, [](Erdos.md) cardinals below any [Ramsey](Ramsey.md "Ramsey") cardinal. {% cite Sharpe2011 %} - The following are equivalent: {% cite Nielsen2018 %} - $κ$ is completely ineffable. - $κ$ is coherent lt;ω$-Ramsey. - $κ$ has the $ω$-[filter property](Filter_property "Filter property") - Every completely ineffable is a stationary limit of lt;ω$-Ramseys. {% cite Nielsen2018 %} - Completely Ramsey cardinals and $ω$-Ramsey cardinals are completely ineffable. {% cite Nielsen2018 %} - $ω$-Ramsey cardinals are limits of completely ineffable cardinals.{% cite Holy2018 %}