In
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
, a branch of
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 ...
, the axiom of uniformization is a weak form of the
axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collectio ...
. It states that if
is a
subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
of
, where
and
are
Polish space
In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named bec ...
s, then there is a subset
of
that is a
partial function
In mathematics, a partial function from a set to a set is a function from a subset of (possibly itself) to . The subset , that is, the domain of viewed as a function, is called the domain of definition of . If equals , that is, if is de ...
from
to
, and whose domain (the
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of all
such that
exists) equals
:
Such a function is called a uniformizing function for
, or a uniformization of
.
To see the relationship with the axiom of choice, observe that
can be thought of as associating, to each element of
, a subset of
. A uniformization of
then picks exactly one element from each such subset, whenever the subset is
non-empty
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other t ...
. Thus, allowing arbitrary sets ''X'' and ''Y'' (rather than just Polish spaces) would make the axiom of uniformization equivalent to the axiom of choice.
A
pointclass
In the mathematical field of descriptive set theory, a pointclass is a collection of sets of points, where a ''point'' is ordinarily understood to be an element of some perfect Polish space. In practice, a pointclass is usually characterized by ...
is said to have the uniformization property if every
relation
Relation or relations may refer to:
General uses
*International relations, the study of interconnection of politics, economics, and law on a global level
*Interpersonal relationship, association or acquaintance between two or more people
*Public ...
in
can be uniformized by a partial function in
. The uniformization property is implied by the
scale property, at least for
adequate pointclass In the mathematical field of descriptive set theory, a pointclass can be called adequate if it contains all recursive
Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in ...
es of a certain form.
It follows from
ZFC alone that
and
have the uniformization property. It follows from the existence of sufficient
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 ...
s that
*
and
have the uniformization property for every
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
.
*Therefore, the collection of
projective set
In the mathematical field of descriptive set theory, a subset A of a Polish space X is projective if it is \boldsymbol^1_n for some positive integer n. Here A is
* \boldsymbol^1_1 if A is analytic
* \boldsymbol^1_n if the complement of A, X\se ...
s has the uniformization property.
*Every relation in
L(R)
In set theory, L(R) (pronounced L of R) is the smallest transitive inner model of ZF containing all the ordinals and all the reals.
Construction
It can be constructed in a manner analogous to the construction of L (that is, Gödel's constructi ...
can be uniformized, but ''not necessarily'' by a function in L(R). In fact, L(R) does not have the uniformization property (equivalently, L(R) does not satisfy the axiom of uniformization).
**(Note: it's trivial that every relation in L(R) can be uniformized ''in V'', assuming V satisfies the axiom of choice. The point is that every such relation can be uniformized in some transitive inner model of V in which the
axiom of determinacy
In mathematics, the axiom of determinacy (abbreviated as AD) is a possible axiom for set theory introduced by Jan Mycielski and Hugo Steinhaus in 1962. It refers to certain two-person topological games of length ω. AD states that every game of ...
holds.)
References
* {{cite book , author=Moschovakis, Yiannis N. , authorlink = Yiannis N. Moschovakis, title=Descriptive Set Theory , url=https://archive.org/details/descriptivesetth0000mosc , url-access=registration , publisher=North Holland , year=1980 , isbn=0-444-70199-0
Set theory
Descriptive set theory
Axiom of choice