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

${\displaystyle \{x\in X\mid \exists y\in Y:(x,y)\in R\}\,}$

Such a function is called a uniformizing function for ${\displaystyle R}$, or a uniformization of ${\displaystyle R}$.

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

It follows from ZFC alone that ${\displaystyle {\boldsymbol {\Pi }}_{1}^{1}}$ and ${\displaystyle {\boldsymbol {\Sigma }}_{2}^{1}}$ have the uniformization property. It follows from the existence of sufficient large cardinals that

• ${\displaystyle {\boldsymbol {\Pi }}_{2n+1}^{1}}$ and ${\displaystyle {\boldsymbol {\Sigma }}_{2n+2}^{1}}$ have the uniformization property for every natural number ${\displaystyle n}$.
• 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.)