Uniformization (set theory)

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In set theory, the axiom of uniformization, a weak form of the axiom of choice, 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 (in the sense of the set of all x such that f(x) exists) equals

\{x\in X|\exists y\in Y (x,y)\in 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 nonempty. Thus, allowing arbitrary sets X and Y (rather than just Polish spaces) would make the axiom of uniformization equivalent to AC.

A pointclass \boldsymbol{\Gamma} is said to have the uniformization property if every relation R in \boldsymbol{\Gamma} can be uniformized by a partial function in \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 \boldsymbol{\Pi}^1_1 and \boldsymbol{\Sigma}^1_2 have the uniformization property. It follows from the existence of sufficient large cardinals that

  • \boldsymbol{\Pi}^1_{2n+1} and \boldsymbol{\Sigma}^1_{2n+2} 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 AC. The point is that every such relation can be uniformized in some transitive inner model of V in which AD holds.)

References[edit]

  • Moschovakis, Yiannis N. (1980). Descriptive Set Theory. North Holland. ISBN 0-444-70199-0.