In the
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
of
transfinite number
In mathematics, transfinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite. These include the transfinite cardinals, which are cardinal numbers used to qua ...
s, an ineffable cardinal is a certain kind of
large cardinal
In the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers. Cardinals with such properties are, as the name suggests, generally very "large" (for example, bigger than the least Î ...
number, introduced by . In the following definitions,
will always be a
regular uncountable
In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal num ...
cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. Th ...
.
A
cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. Th ...
is called almost ineffable if for every
(where
is the
powerset
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is postu ...
of
) with the property that
is a subset of
for all ordinals
, there is a subset
of
having cardinality
and
homogeneous
Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
for
, in the sense that for any
in
,
.
A
cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. Th ...
is called ineffable if for every binary-valued function
, there is a
stationary subset of
on which
is
homogeneous
Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
: 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
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
such that each ,
there is such that is stationary in .
More generally,
is called
-ineffable (for a positive integer
) if for every
there is a stationary subset of
on which
is
-
homogeneous
Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
(takes the same value for all unordered
-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
-ineffable for every
. If
is
-ineffable, then the set of
-ineffable cardinals below
is a stationary subset of
.
Every ''n''-ineffable cardinal is ''n''-almost ineffable (with set of ''n''-almost ineffable below it stationary), and every ''n''-almost ineffable is ''n''-subtle (with set of ''n''-subtle below it stationary). The least ''n''-subtle cardinal is not even
weakly compact (and unlike ineffable cardinals, the least ''n''-almost ineffable is
-describable), but ''n''-1-ineffable cardinals are stationary below every ''n''-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 ''n'' > 1 in place of 2 would lead to the same definition, so completely ineffable cardinals are totally ineffable (and have greater
consistency strength
In mathematical logic, two theories are equiconsistent if the consistency of one theory implies the consistency of the other theory, and vice versa. In this case, they are, roughly speaking, "as consistent as each other".
In general, it is not po ...
). 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
here.
See also
*
List of large cardinal properties
This page includes a list of cardinals with large cardinal properties. It is arranged roughly in order of the consistency strength of the axiom asserting the existence of cardinals with the given property. Existence of a cardinal number κ of a g ...
References
*.
*
Large cardinals
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