Ineffable cardinal
Kind of large cardinal number / From Wikipedia, the free encyclopedia
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, will always be a regular uncountable cardinal number.
A cardinal number is called almost ineffable if for every
(where
is the powerset of
) 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 ⟨Aα : α ∈ κ⟩ such that each Aα ⊆ α,
there is A ⊆ κ such that {α ∈ κ : A ∩ α = Aα} is stationary in κ.
Another equivalent formulation is that a regular uncountable cardinal is ineffable if for every set
of cardinality
of subsets of
, there is a normal (i.e. closed under diagonal intersection) non-trivial
-complete filter
on
deciding
: that is, for any
, either
or
.[1] This is similar to a characterization of weakly compact cardinals.
More generally, is called
-ineffable (for a positive integer
) if for every
there is a stationary subset of
on which
is
-homogeneous (takes the same value for all unordered
-tuples drawn from the subset). Thus, it is ineffable if and only if it is 2-ineffable. Ineffability is strictly weaker than 3-ineffability.[2]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 (with set of
-almost ineffable below it stationary), and every
-almost ineffable is
-subtle (with set of
-subtle below it stationary). The least
-subtle cardinal is not even weakly compact (and unlike ineffable cardinals, the least
-almost ineffable is
-describable), 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 (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.