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 [math]\displaystyle{ R }[/math] is a subset of [math]\displaystyle{ X\times Y }[/math], where [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] are Polish spaces, then there is a subset [math]\displaystyle{ f }[/math] of [math]\displaystyle{ R }[/math] that is a partial function from [math]\displaystyle{ X }[/math] to [math]\displaystyle{ Y }[/math], and whose domain (the set of all [math]\displaystyle{ x }[/math] such that [math]\displaystyle{ f(x) }[/math] exists) equals

[math]\displaystyle{ \\{x \in X \mid \exists y \in Y: (x,y) \in R\\}\, }[/math]

Such a function is called a uniformizing function for [math]\displaystyle{ R }[/math], or a uniformization of [math]\displaystyle{ R }[/math].

Uniformization of relation R (light blue) by function f (red).

To see the relationship with the axiom of choice, observe that [math]\displaystyle{ R }[/math] can be thought of as associating, to each element of [math]\displaystyle{ X }[/math], a subset of [math]\displaystyle{ Y }[/math]. A uniformization of [math]\displaystyle{ R }[/math] 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 [math]\displaystyle{ \boldsymbol{\Gamma} }[/math] is said to have the uniformization property if every relation [math]\displaystyle{ R }[/math] in [math]\displaystyle{ \boldsymbol{\Gamma} }[/math] can be uniformized by a partial function in [math]\displaystyle{ \boldsymbol{\Gamma} }[/math]. The uniformization property is implied by the scale property, at least for adequate pointclasses of a certain form.

It follows from ZFC alone that [math]\displaystyle{ \boldsymbol{\Pi}^1_1 }[/math] and [math]\displaystyle{ \boldsymbol{\Sigma}^1_2 }[/math] have the uniformization property. It follows from the existence of sufficient large cardinals that

* [math]\displaystyle{ \boldsymbol{\Pi}^1_{2n+1} }[/math] and [math]\displaystyle{ \boldsymbol{\Sigma}^1_{2n+2} }[/math] have the uniformization property for every natural number [math]\displaystyle{ n }[/math].
* 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. (1980). Descriptive Set Theory. North Holland. ISBN 0-444-70199-0. https://archive.org/details/descriptivesetth0000mosc.

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