Ineffable cardinal

From Cantor's Attic

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 $S=\{\alpha<\kappa\mid A\cap \kappa=A_\alpha\}$ 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 1 Ineffable cardinals and the constructible universe 2 Relations with other large cardinals 3 Weakly ineffable cardinal 4 Subtle cardinal 5 Completely ineffable cardinal 6 $n$-ineffable cardinal 7 References

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 $S=\{\alpha<\kappa\mid A\cap \kappa=A_\alpha\}$ 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.
• $\alpha$-Erdős cardinals are subtle. [1]

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]

$n$-ineffable cardinal

The $n$-ineffable cardinals for $2\leq n<\omega$ were introduced by Baumgartner in [4] as a strengthening of ineffable cardinals. A cardinal is $n$-ineffable if for every function $F:[\kappa]^2\rightarrow n$ there is a stationary $H\subseteq\kappa$ such that $F\upharpoonright [H]^n$ is constant. A cardinal $\kappa$ is totally ineffable if it is $n$-ineffable for every $n$.

• $2$-ineffable cardinals are exactly the ineffable cardinals.
• an $n+1$-ineffable cardinal is a stationary limit of $n$-ineffable cardinals. [4]
• a $1$-iterable cardinal is a stationary limit of totally ineffable cardinals. (this follows from material in [5])

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, Berlin, 2003. 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. 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

5. Gitman, Victoria. Ramsey-like cardinals. The Journal of Symbolic Logic 76(2):519-540, 2011. www   arχiv   MR   bibtex

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