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 {\displaystyle R} is a subset of X × Y {\displaystyle X\times Y} , where X {\displaystyle X} and Y {\displaystyle Y} are Polish spaces, then there is a subset f {\displaystyle f} of R {\displaystyle R} that is a partial function from X {\displaystyle X} to Y {\displaystyle Y} , and whose domain (the set of all x {\displaystyle x} such that f ( x ) {\displaystyle f(x)} exists) equals
{ x ∈ X ∣ ∃ y ∈ Y : ( x , y ) ∈ R } {\displaystyle \{x\in X\mid \exists y\in Y:(x,y)\in R\}\,}
Such a function is called a uniformizing function for R {\displaystyle R} , or a uniformization of R {\displaystyle R} .
To see the relationship with the axiom of choice, observe that R {\displaystyle R} can be thought of as associating, to each element of X {\displaystyle X} , a subset of Y {\displaystyle Y} . A uniformization of R {\displaystyle 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 {\displaystyle {\boldsymbol {\Gamma }}} is said to have the uniformization property if every relation R {\displaystyle R} in Γ {\displaystyle {\boldsymbol {\Gamma }}} can be uniformized by a partial function in Γ {\displaystyle {\boldsymbol {\Gamma }}} . The uniformization property is implied by the scale property, at least for adequate pointclasses of a certain form. It follows from ZFC alone that Π 1 1 {\displaystyle {\boldsymbol {\Pi }}_{1}^{1}} and Σ 2 1 {\displaystyle {\boldsymbol {\Sigma }}_{2}^{1}} have the uniformization property. It follows from the existence of sufficient large cardinals that
Π 2 n + 1 1 {\displaystyle {\boldsymbol {\Pi }}_{2n+1}^{1}} and Σ 2 n + 2 1 {\displaystyle {\boldsymbol {\Sigma }}_{2n+2}^{1}} have the uniformization property for every natural number n {\displaystyle 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.
