Uniformization (set theory)

In set theory, a branch of mathematics, the axiom of uniformization is a weak form of the axiom of choice. It states that if $R$ is a subset of $X\times Y$, where $X$ and $Y$ are Polish spaces, then there is a subset $f$ of $R$ that is a partial function from $X$ to $Y$, and whose domain (the set of all $x$ such that $f(x)$ exists) equals
: $\\,$
Such a function is called a uniformizing function for $R$, or a uniformization of $R$.

To see the relationship with the axiom of choice, observe that $R$ can be thought of as associating, to each element of $X$, a subset of $Y$. A uniformization of $R$ 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.

A pointclass $\boldsymbol$ is said to have the uniformization property if every relation $R$ in $\boldsymbol$ can be uniformized by a partial function in $\boldsymbol$. The uniformization property is implied by the scale property, at least for adequate pointclasses of a certain form.

It follows from ZFC alone that $\boldsymbol^1_1$ and $\boldsymbol^1_2$ have the uniformization property. It follows from the existence of sufficient large cardinals that
*$\boldsymbol^1_$ and $\boldsymbol^1_$ have the uniformization property for every natural number $n$.
*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

* {{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