In the mathematics of transfinite numbers, an ineffable cardinal is a certain kind of large cardinal number, introduced by Jensen & Kunen (1969).
will always be a regular uncountable cardinal number.
A cardinal number
is called almost ineffable if for every
A cardinal number
is called ineffable if for every binary-valued function
, there is a stationary subset of
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
: α ∈ κ ⟩
{ α ∈ κ :
Another equivalent formulation is that a regular uncountable cardinal
is ineffable if for every set
, there is a normal (i.e. closed under diagonal intersection) non-trivial
[1] This is similar to a characterization of weakly compact cardinals.
-ineffable (for a positive integer
-homogeneous (takes the same value for all unordered
-tuples drawn from the subset).
Ineffability is strictly weaker than 3-ineffability.[2]p.
-almost ineffable (with set of
-almost ineffable below it stationary), and every
-subtle cardinal is not even weakly compact (and unlike ineffable cardinals, the least
-ineffable cardinals are stationary below every
A cardinal κ is completely ineffable if there is a non-empty
> 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 in the section below.