Ineffable cardinal

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Ineffable cardinals were introduced by Jensen and Kunen in [1] 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 [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 $|\alpha|$.


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.
  • $\omega$-Erdős cardinals are stationary limits of ineffable cardinals, but not ineffable since they are $\Pi_1^1$-describable. [2]
  • 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 $S=\{\alpha<\kappa\mid A\cap \alpha=A_\alpha\}$ 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 stationary limits of totally indescribable cardinals. [1, 4]
  • The least subtle cardinal is not weakly compact as it is $\Pi_1^1$-describable.
  • $\alpha$-Erdős cardinals are subtle. [1]
  • If $δ$ is a subtle cardinal,
    • the set of cardinals $κ$ below $δ$ that are strongly uplifting in $V_δ$ is stationary.[5]
    • 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.”}$.[6] (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 $\eta < δ$.[6]

Ethereal cardinal

To be expanded.

$n$-ineffable cardinal

The $n$-ineffable cardinals for $2\leq n<\omega$ were introduced by Baumgartner in [7] 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. [7]

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 [8])

Helix

(Information in this subsection come from [4] 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 $f(A) < \min(A)$.
  • $E$ 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.[9]

Completely ineffable cardinal

Completely ineffable cardinals were introduced in [3] 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.

  • Completely ineffable cardinals are downward absolute to $L$. [3]
  • Completely ineffable cardinals are limits of ineffable cardinals. [3]

References

  1. Jensen, Ronald and Kunen, Kenneth. Some combinatorial properties of $L$ and $V$. Unpublished, 1969. www   bibtex
  2. Jech, Thomas J. Set Theory. Third, Springer-Verlag, Berlin, 2003. (The third millennium edition, revised and expanded) www   bibtex
  3. 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
  4. Friedman, Harvey M. Subtle cardinals and linear orderings. , 1998. www   bibtex
  5. Hamkins, Joel David and Johnstone, Thomas A. Strongly uplifting cardinals and the boldface resurrection axioms. , 2014. www   arχiv   bibtex
  6. Rathjen, Michael. The art of ordinal analysis. , 2006. www   bibtex
  7. Baumgartner, James. Ineffability properties of cardinals. I. 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, Amsterdam, 1975. MR   bibtex
  8. Gitman, Victoria. Ramsey-like cardinals. The Journal of Symbolic Logic 76(2):519-540, 2011. www   arχiv   MR   bibtex
  9. Kentaro, Sato. Double helix in large large cardinals and iteration ofelementary embeddings. , 2007. www   bibtex
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