- Source: Ineffable cardinal
In the mathematics of transfinite numbers, an ineffable cardinal is a certain kind of large cardinal number, introduced by Jensen & Kunen (1969). In the following definitions,
κ
{\displaystyle \kappa }
will always be a regular uncountable cardinal number.
A cardinal number
κ
{\displaystyle \kappa }
is called almost ineffable if for every
f
:
κ
→
P
(
κ
)
{\displaystyle f:\kappa \to {\mathcal {P}}(\kappa )}
(where
P
(
κ
)
{\displaystyle {\mathcal {P}}(\kappa )}
is the powerset of
κ
{\displaystyle \kappa }
) with the property that
f
(
δ
)
{\displaystyle f(\delta )}
is a subset of
δ
{\displaystyle \delta }
for all ordinals
δ
<
κ
{\displaystyle \delta <\kappa }
, there is a subset
S
{\displaystyle S}
of
κ
{\displaystyle \kappa }
having cardinality
κ
{\displaystyle \kappa }
and homogeneous for
f
{\displaystyle f}
, in the sense that for any
δ
1
<
δ
2
{\displaystyle \delta _{1}<\delta _{2}}
in
S
{\displaystyle S}
,
f
(
δ
1
)
=
f
(
δ
2
)
∩
δ
1
{\displaystyle f(\delta _{1})=f(\delta _{2})\cap \delta _{1}}
.
A cardinal number
κ
{\displaystyle \kappa }
is called ineffable if for every binary-valued function
f
:
[
κ
]
2
→
{
0
,
1
}
{\displaystyle f:[\kappa ]^{2}\to \{0,1\}}
, there is a stationary subset of
κ
{\displaystyle \kappa }
on which
f
{\displaystyle f}
is homogeneous: that is, either
f
{\displaystyle f}
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
κ
{\displaystyle \kappa }
is ineffable if for every sequence
⟨
A
α
:
α
∈
κ
⟩
{\displaystyle \langle A_{\alpha }:\alpha \in \kappa \rangle }
such that each
A
α
⊆
α
{\displaystyle A_{\alpha }\subseteq \alpha }
,
there is
A
⊆
κ
{\displaystyle A\subseteq \kappa }
such that
{
α
∈
κ
:
A
∩
α
=
A
α
}
{\displaystyle \{\alpha \in \kappa :A\cap \alpha =A_{\alpha }\}}
is stationary in κ.
Another equivalent formulation is that a regular uncountable cardinal
κ
{\displaystyle \kappa }
is ineffable if for every set
S
{\displaystyle S}
of cardinality
κ
{\displaystyle \kappa }
of subsets of
κ
{\displaystyle \kappa }
, there is a normal (i.e. closed under diagonal intersection) non-trivial
κ
{\displaystyle \kappa }
-complete filter
F
{\displaystyle {\mathcal {F}}}
on
κ
{\displaystyle \kappa }
deciding
S
{\displaystyle S}
: that is, for any
X
∈
S
{\displaystyle X\in S}
, either
X
∈
F
{\displaystyle X\in {\mathcal {F}}}
or
κ
∖
X
∈
F
{\displaystyle \kappa \setminus X\in {\mathcal {F}}}
. This is similar to a characterization of weakly compact cardinals.
More generally,
κ
{\displaystyle \kappa }
is called
n
{\displaystyle n}
-ineffable (for a positive integer
n
{\displaystyle n}
) if for every
f
:
[
κ
]
n
→
{
0
,
1
}
{\displaystyle f:[\kappa ]^{n}\to \{0,1\}}
there is a stationary subset of
κ
{\displaystyle \kappa }
on which
f
{\displaystyle f}
is
n
{\displaystyle n}
-homogeneous (takes the same value for all unordered
n
{\displaystyle n}
-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 cardinal is a cardinal that is
n
{\displaystyle n}
-ineffable for every
2
≤
n
<
ℵ
0
{\displaystyle 2\leq n<\aleph _{0}}
. If
κ
{\displaystyle \kappa }
is
(
n
+
1
)
{\displaystyle (n+1)}
-ineffable, then the set of
n
{\displaystyle n}
-ineffable cardinals below
κ
{\displaystyle \kappa }
is a stationary subset of
κ
{\displaystyle \kappa }
.
Every
n
{\displaystyle n}
-ineffable cardinal is
n
{\displaystyle n}
-almost ineffable (with set of
n
{\displaystyle n}
-almost ineffable below it stationary), and every
n
{\displaystyle n}
-almost ineffable is
n
{\displaystyle n}
-subtle (with set of
n
{\displaystyle n}
-subtle below it stationary). The least
n
{\displaystyle n}
-subtle cardinal is not even weakly compact (and unlike ineffable cardinals, the least
n
{\displaystyle n}
-almost ineffable is
Π
2
1
{\displaystyle \Pi _{2}^{1}}
-describable), but
(
n
−
1
)
{\displaystyle (n-1)}
-ineffable cardinals are stationary below every
n
{\displaystyle n}
-subtle cardinal.
A cardinal κ is completely ineffable if there is a non-empty
R
⊆
P
(
κ
)
{\displaystyle R\subseteq {\mathcal {P}}(\kappa )}
such that
- every
A
∈
R
{\displaystyle A\in R}
is stationary
- for every
A
∈
R
{\displaystyle A\in R}
and
f
:
[
κ
]
2
→
{
0
,
1
}
{\displaystyle f:[\kappa ]^{2}\to \{0,1\}}
, there is
B
⊆
A
{\displaystyle B\subseteq A}
homogeneous for f with
B
∈
R
{\displaystyle B\in R}
.
Using any finite
n
{\displaystyle 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
Π
n
1
{\displaystyle \Pi _{n}^{1}}
-indescribable for every n, but the property of being completely ineffable is
Δ
1
2
{\displaystyle \Delta _{1}^{2}}
.
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
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