Uniformization (set theory)

In <a href="/facts/Set_theory/1ptfsvUP">set theory</a>, a branch of <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the axiom of uniformization is a weak form of the <a href="/facts/Axiom_of_choice/1Ura2HTh">axiom of choice</a>. It states that if 
 
 
 
 R
 
 
 {\displaystyle R}
 
 is a <a href="/facts/Subset/SJGERmbA">subset</a> of 
 
 
 
 X
 ×
 Y
 
 
 {\displaystyle X\times Y}
 
, where 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 are <a href="/facts/Polish_space/CkUFcGap">Polish spaces</a>, then there is a subset 
 
 
 
 f
 
 
 {\displaystyle f}
 
 of 
 
 
 
 R
 
 
 {\displaystyle R}
 
 that is a <a href="/facts/Partial_function/0vJMw0Nm">partial function</a> from 
 
 
 
 X
 
 
 {\displaystyle X}
 
 to 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
, and whose domain (the <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> 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 <a href="/facts/Empty_set/ffEk3eA6">non-empty</a>. 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 <a href="/facts/Pointclass/k6fMNJa7">pointclass</a> 
 
 
 
 
 Γ
 
 
 
 {\displaystyle {\boldsymbol {\Gamma }}}
 
 is said to have the uniformization property if every <a href="/facts/Binary_relation/r9BwGgaK">relation</a> 
 
 
 
 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 <a href="/facts/Scale_property/5vNIpNFS">scale property</a>, at least for <a href="/facts/Adequate_pointclass/0z7xeXwV">adequate pointclasses</a> of a certain form.
It follows from <a href="/facts/ZFC/UYIsPbXw">ZFC</a> 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 <a href="/facts/Large_cardinal/jkSeV92D">large cardinals</a> that

<ul><li>
 
 
 
 
 
 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 <a href="/facts/Natural_number/u65og8JE">natural number</a> 
 
 
 
 n
 
 
 {\displaystyle n}
 
.</li>
<li>Therefore, the collection of <a href="/facts/Projective_set/Ip9qupJn">projective sets</a> has the uniformization property.</li>
<li>Every relation in <a href="/facts/L(R)/wDtAnbll">L(R)</a> 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).
<ul><li>(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 <a href="/facts/Axiom_of_determinacy/yztrbXJY">axiom of determinacy</a> holds.)</li></ul></li></ul>

<ul><li><a href="/facts/Yiannis_N._Moschovakis/ajc4e46V">Moschovakis, Yiannis N.</a> (1980). <a href="https://archive.org/details/descriptivesetth0000mosc">Descriptive Set Theory</a>. North Holland. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-444-70199-0.</li></ul>

Uniformization (set theory) open-in-new

Uniformization (set theory)