- Source: 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
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.
Kata Kunci Pencarian:
- Uniformization (set theory)
- List of set theory topics
- Uniformization
- List of mathematical logic topics
- Constructive set theory
- Scale (descriptive set theory)
- Glossary of set theory
- Named set theory
- Causal sets
- L(R)
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