Ineffable cardinal

• Ineffable cardinals and the constructible universe
• Relations with other large cardinals
• Weakly ineffable cardinal
• Subtle cardinal
• Ethereal cardinal
• $n$-ineffable cardinal
  • Helix
• Completely ineffable cardinal

Ineffable cardinals were introduced by Jensen and Kunen in (Jensen & Kunen, 1969) and arose out of their study of $\diamondsuit$ principles. 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 (Jensen & Kunen, 1969). 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 (Jensen & Kunen, 1969). 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 iff it is ineffable. (Holy & Schlicht, 2018)

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. (Jensen & Kunen, 1969)

If $0^\sharp$ exists, then every Silver indiscernible is ineffable in $L$. (Jech, 2003)

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. (Holy & Schlicht, 2018; Gitman, 2011)

Relations with other large cardinals

• Measurable cardinals are ineffable and stationary limits of ineffable cardinals.
• $\omega$-Erdős cardinals are stationary limits of ineffable cardinals, but not ineffable since they are $\Pi_1^1$-describable. (Jech, 2003)
• Ineffable cardinals are $\Pi^1_2$-indescribable (Jensen & Kunen, 1969).
• Ineffable cardinals are limits of totally indescribable cardinals. ((Jensen & Kunen, 1969; Abramson et al., 1977) for proof)
• For a cardinal $κ=κ^{<κ}$, $κ$ is ineffable iff it is normal 0-Ramsey. (Nielsen & Welch, 2018)

Weakly ineffable cardinal

Weakly ineffable cardinals (also called almost ineffable) were introduced by Jensen and Kunen in (Jensen & Kunen, 1969) 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 $L$. (Jensen & Kunen, 1969)
• Weakly ineffable cardinals are $\Pi_1^1$-indescribable. (Jensen & Kunen, 1969)
• Ineffable cardinals are limits of weakly ineffable cardinals.
• Weakly ineffable cardinals are limits of totally indescribable cardinals. (Jensen & Kunen, 1969) ((Abramson et al., 1977) for proof)
• For a cardinal $κ=κ^{<κ}$, $κ$ is weakly ineffable iff it is genuine 0-Ramsey. (Nielsen & Welch, 2018)

Subtle cardinal

Subtle cardinals were introduced by Jensen and Kunen in (Jensen & Kunen, 1969) 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 $L$. (Jensen & Kunen, 1969)
• Weakly ineffable cardinals are limits of subtle cardinals. (Jensen & Kunen, 1969)
• Subtle cardinals are stationary limits of totally indescribable cardinals. (Jensen & Kunen, 1969; Friedman, 1998)
• The least subtle cardinal is not weakly compact as it is $\Pi_1^1$-describable.
• $\alpha$-Erdős cardinals are subtle. (Jensen & Kunen, 1969)
• If $δ$ is a subtle cardinal,
  • the set of cardinals $κ$ below $δ$ that are strongly uplifting in $V_δ$ is stationary. (Hamkins & Johnstone, 2014)
  • 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.”}$. (Rathjen, 2006) (The set of cardinals $κ$ below $δ$ that are $\mathcal{A}$-shrewd in $V_δ$ is stationary.)
  • there is an $\eta$-shrewd cardinal below $δ$ for all $\eta < δ$. (Rathjen, 2006)

Ethereal cardinal

To be expanded.

$n$-ineffable cardinal

The $n$-ineffable cardinals for $2\leq n<\omega$ were introduced by Baumgartner in (Baumgartner, 1975) 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. (Baumgartner, 1975)

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 (Gitman, 2011))

Helix

(Information in this subsection come from (Friedman, 1998) 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 double helix of $n$-fold variants and earlier known although smaller. (Kentaro, 2007)

Completely ineffable cardinal

Completely ineffable cardinals were introduced in (Abramson et al., 1977) 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 $L$. (Abramson et al., 1977)
• Completely ineffable cardinals are limits of ineffable cardinals. (Abramson et al., 1977)
• There are stationarily many completely ineffable, greatly Erdős cardinals below any Ramsey cardinal. (Sharpe & Welch, 2011)
• The following are equivalent: (Nielsen & Welch, 2018)
  • $κ$ is completely ineffable.
  • $κ$ is coherent $<ω$-Ramsey.
  • $κ$ has the $ω$-filter property
• Every completely ineffable is a stationary limit of $<ω$-Ramseys. (Nielsen & Welch, 2018)
• Completely Ramsey cardinals and $ω$-Ramsey cardinals are completely ineffable. (Nielsen & Welch, 2018)
• $ω$-Ramsey cardinals are limits of completely ineffable cardinals.(Holy & Schlicht, 2018)

References

1. Jensen, R., & Kunen, K. (1969). Some combinatorial properties of L and V. http://www.mathematik.hu-berlin.de/ raesch/org/jensen.html
2. Holy, P., & Schlicht, P. (2018). A hierarchy of Ramsey-like cardinals. Fundamenta Mathematicae, 242, 49–74. https://doi.org/10.4064/fm396-9-2017
3. Jech, T. J. (2003). Set Theory (Third). Springer-Verlag. https://logic.wikischolars.columbia.edu/file/view/Jech%2C+T.+J.+%282003%29.+Set+Theory+%28The+3rd+millennium+ed.%29.pdf
4. Gitman, V. (2011). Ramsey-like cardinals. The Journal of Symbolic Logic, 76(2), 519–540. http://boolesrings.org/victoriagitman/files/2011/08/ramseylikecardinals.pdf
5. Abramson, F., Harrington, L., Kleinberg, E., & Zwicker, W. (1977). Flipping properties: a unifying thread in the theory of large cardinals. Ann. Math. Logic, 12(1), 25–58.
6. Nielsen, D. S., & Welch, P. (2018). Games and Ramsey-like cardinals.
7. Friedman, H. M. (1998). Subtle cardinals and linear orderings. https://u.osu.edu/friedman.8/files/2014/01/subtlecardinals-1tod0i8.pdf
8. Hamkins, J. D., & Johnstone, T. A. (2014). Strongly uplifting cardinals and the boldface resurrection axioms.
9. Rathjen, M. (2006). The art of ordinal analysis. http://www.icm2006.org/proceedings/Vol_II/contents/ICM_Vol_2_03.pdf
10. Baumgartner, J. (1975). Ineffability properties of cardinals. I. In Infinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erdős on his 60th birthday), Vol. I (pp. 109–130. Colloq. Math. Soc. János Bolyai, Vol. 10). North-Holland.
11. Kentaro, S. (2007). Double helix in large large cardinals and iteration of elementary embeddings. Annals of Pure and Applied Logic, 146(2-3), 199–236. https://doi.org/10.1016/j.apal.2007.02.003
12. Sharpe, I., & Welch, P. (2011). Greatly Erdős cardinals with some generalizations to the Chang and Ramsey properties. Ann. Pure Appl. Logic, 162(11), 863–902. https://doi.org/10.1016/j.apal.2011.04.002

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