Uniformization (set theory) explained
In set theory, a branch of mathematics, the axiom of uniformization is a weak form of the axiom of choice. It states that if
is a
subset of
, where
and
are
Polish spaces, then there is a subset
of
that is a
partial function from
to
, and whose domain (the
set of all
such that
exists) equals
\{x\inX\mid\existsy\inY:(x,y)\inR\}
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. Thus, allowing arbitrary sets
X and
Y (rather than just Polish spaces) would make the axiom of uniformization equivalent to the axiom of choice.
is said to have the
uniformization property if every
relation
in
can be uniformized by a partial function in
. The uniformization property is implied by the
scale property, at least for
adequate pointclasses of a certain form.
It follows from ZFC alone that
and
have the uniformization property. It follows from the existence of sufficient
large cardinals that
and
have the uniformization property for every
natural number
.
- Therefore, the collection of projective sets has the uniformization property.
- Every relation in L(R) 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 holds.)
References
. Moschovakis, Yiannis N. . Yiannis N. Moschovakis. Descriptive Set Theory . registration . North Holland . 1980 . 0-444-70199-0.
