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, will always be a regular uncountable cardinal number.
A cardinal number is called almost ineffable if for every 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. An equivalent formulation is that a cardinal is ineffable if for every sequence such that each,
there is such that is stationary in.
Another equivalent formulation is that a regular uncountable cardinal is ineffable if for every set of cardinality of subsets of, there is a normal non-trivial -complete filter on deciding : that is, for any, either or. This is similar to a characterization of weakly compact cardinals.
More generally, is called -ineffable if for every there is a stationary subset of on which is homogeneous. Thus, it is ineffable if and only if it is 2-ineffable. Ineffability is strictly weaker than 3-ineffability.p. 399
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 -ineffable cardinal is -almost ineffable, and every -almost ineffable is -subtle. The least -subtle cardinal is not even weakly compact, but -ineffable cardinals are stationary below every -subtle cardinal.
A cardinal κ is completely ineffable if there is a non-empty such that
- every is stationary
- for every and, there is homogeneous for f with.
Using any finite > 1 in place of 2 would lead to the same definition, so completely ineffable cardinals are totally ineffable. 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 in the section below.