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
κ
is called almost ineffable if for every
f
:
κ
→
P
(
κ
)
(where
P
(
κ
)
is the powerset of
κ
) with the property that
f
(
δ
)
is a subset of
δ
for all ordinals
δ
<
κ
, there is a subset
S
of
κ
having cardinal
κ
and homogeneous for
f
, in the sense that for any
δ
1
<
δ
2
in
S
,
f
(
δ
1
)
=
f
(
δ
2
)
∩
δ
1
.
A cardinal number
κ
is called ineffable if for every binary-valued function
f
:
[
κ
]
2
→
{
0
,
1
}
, there is a stationary subset of
κ
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.
More generally,
κ
is called
n
-ineffable (for a positive integer
n
) if for every
f
:
[
κ
]
n
→
{
0
,
1
}
there is a stationary subset of
κ
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
≤
n
<
ℵ
0
. If
κ
is
(
n
+
1
)
-ineffable, then the set of
n
-ineffable cardinals below
κ
is a stationary subset of
κ
.
Totally ineffable cardinals are of greater consistency strength than subtle cardinals and of lesser consistency strength than remarkable cardinals. A list of large cardinal axioms by consistency strength is available here.
