# Ineffable cardinal

In the mathematics of transfinite numbers, an **ineffable cardinal** is a certain kind of large cardinal number, introduced by Jensen & Kunen (1969).

A cardinal number is called **almost ineffable** if for every (where is the powerset of ) with the property that is a subset of for all ordinals , there is a subset of having cardinality and homogeneous for , in the sense that for any in , .

A cardinal number is called **ineffable** if for every binary-valued function , there is a stationary subset of on which is homogeneous: that is, either maps all unordered pairs of elements drawn from that subset to zero, or it maps all such unordered pairs to one.

More generally, is called **-ineffable** (for a positive integer ) if for every there is a stationary subset of on which is **-homogeneous** (takes the same value for all unordered -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 -ineffable for every . If is -ineffable, then the set of -ineffable cardinals below is a stationary subset of .

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 -describable), but *n*-1-ineffable cardinals are stationary below every *n*-subtle cardinal.

A cardinal κ is **completely ineffable** iff there is a non-empty such that

- every is stationary

- for every and , there is homogeneous for *f* with .

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 -indescribable for every *n*, but the property of being completely ineffable is .

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.

## References

- Friedman, Harvey (2001), "Subtle cardinals and linear orderings",
*Annals of Pure and Applied Logic*,**107**(1–3): 1–34, doi:10.1016/S0168-0072(00)00019-1. - Jensen, Ronald; Kunen, Kenneth (1969),
*Some Combinatorial Properties of L and V*, Unpublished manuscript