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Uniformization (set theory)

释义

References

In set theory, the axiom of uniformization, a weak form of the axiom of choice, states that if is a subset of , where and are Polish spaces,

then there is a subset of that is a partial function from to , and whose domain (in the sense of the set of all such that exists) equals

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

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

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

and have the uniformization property for every natural number .
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

{{cite book | author=Moschovakis, Yiannis N. | title=Descriptive Set Theory | publisher=North Holland | year=1980 |isbn=0-444-70199-0}}

3 : Set theory|Descriptive set theory|Axiom of choice

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