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×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

{x∈X∣∃y∈Y:(x,y)∈R}

Such a function is called a uniformizing function for R, or a uniformization of R.

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

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

It follows from ZFC alone that Π11 and Σ21 have the uniformization property. It follows from the existence of sufficient large cardinals that

* Π2n+11 and Σ2n+21 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

* Moschovakis, Yiannis N. (1980). Descriptive Set Theory. North Holland. ISBN 0-444-70199-0. https://archive.org/details/descriptivesetth0000mosc.

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