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

A cardinal number is called almost ineffable is the powerset and homogeneous.

A cardinal number is called ineffable, there is a stationary subset is homogeneous is ineffable if for every sequence is stationary in .

, there is a normal (i.e. closed under diagonal intersection-complete filter. This is similar to a characterization of weakly compact cardinals.

is called -ineffable-tuples drawn from the subset). Thus, it is ineffable if and only if it is 2-ineffable. Ineffability is strictly weaker than 3-ineffability.p. 399

A totally ineffable.

-subtle-describable-subtle cardinal.

A cardinal κ is completely ineffable such that

• is stationary

• homogeneous for f.

>1 in place of 2 would lead to the same definition, so completely ineffable cardinals are totally ineffable (and have greater consistency strength-indescribable for every n.

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.

See also

• List of large cardinal properties

References

• .

Citations