Ineffable Cardinal

In the mathematics of transfinite numbers, an ineffable cardinal is a certain kind of large cardinal number, introduced by . In the following definitions, $\kappa$ will always be a regular uncountable cardinal number.

A cardinal number $\kappa$ is called almost ineffable if for every $f: \kappa \to \mathcal(\kappa)$ (where $\mathcal(\kappa)$ is the powerset of $\kappa$) with the property that $f(\delta)$ is a subset of $\delta$ for all ordinals $\delta < \kappa$, there is a subset $S$ of $\kappa$ having cardinality $\kappa$ and homogeneous for $f$, in the sense that for any $\delta_1 < \delta_2$ in $S$, $f(\delta_1) = f(\delta_2) \cap \delta_1$.

A cardinal number $\kappa$ is called ineffable if for every binary-valued function $f :$kappa2\to \, there is a stationary subset of $\kappa$ on which $f$ is homogeneous: that is, either $f$ maps all unordered pairs of elements drawn from that subset to zero, or it maps all such unordered pairs to one. An equivalent formulation is that a cardinal $\kappa$ is ineffable if for every sequence such that each ,
there is such that is stationary in .

More generally, $\kappa$ is called $n$-ineffable (for a positive integer $n$) if for every $f :$kappan\to \ there is a stationary subset of $\kappa$ on which $f$ is $n$-homogeneous (takes the same value for all unordered $n$-tuples drawn from the subset). Thus, it is ineffable if and only if it is 2-ineffable.

A totally ineffable cardinal is a cardinal that is $n$-ineffable for every $2 \leq n < \aleph_0$. If $\kappa$ is $(n+1)$-ineffable, then the set of $n$-ineffable cardinals below $\kappa$ is a stationary subset of $\kappa$.

Every ''n''-ineffable cardinal is ''n''-almost ineffable (with set of ''n''-almost ineffable below it stationary), and every ''n''-almost ineffable is ''n''-subtle (with set of ''n''-subtle below it stationary). The least ''n''-subtle cardinal is not even weakly compact (and unlike ineffable cardinals, the least ''n''-almost ineffable is $\Pi^1_2$-describable), but ''n''-1-ineffable cardinals are stationary below every ''n''-subtle cardinal.

A cardinal κ is completely ineffable if there is a non-empty $R \subseteq \mathcal(\kappa)$ such that

- every $A \in R$ is stationary

- for every $A \in R$ and $f :$kappa2\to \, there is $B \subseteq A$ homogeneous for ''f'' with $B \in R$.

Using any finite ''n'' > 1 in place of 2 would lead to the same definition, so completely ineffable cardinals are totally ineffable (and have greater consistency strength). Completely ineffable cardinals are $\Pi^1_n$-indescribable for every ''n'', but the property of being completely ineffable is $\Delta^2_1$.

The consistency strength of completely ineffable is below that of 1-iterable cardinals, which in turn is below remarkable cardinals, which in turn is below ω-Erdős cardinals. A list of large cardinal axioms by consistency strength is available here.

See also

* List of large cardinal properties

References

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Large cardinals

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