- Source: Difference hierarchy
In set theory, a branch of mathematics, the difference hierarchy over a pointclass is a hierarchy of larger pointclasses
generated by taking differences of sets. If Γ is a pointclass, then the set of differences in Γ is
{
A
:
∃
C
,
D
∈
Γ
(
A
=
C
∖
D
)
}
{\displaystyle \{A:\exists C,D\in \Gamma (A=C\setminus D)\}}
. In usual notation, this set is denoted by 2-Γ. The next level of the hierarchy is denoted by 3-Γ and consists of differences of three sets:
{
A
:
∃
C
,
D
,
E
∈
Γ
(
A
=
C
∖
(
D
∖
E
)
)
}
{\displaystyle \{A:\exists C,D,E\in \Gamma (A=C\setminus (D\setminus E))\}}
. This definition can be extended recursively into the transfinite to α-Γ for some ordinal α.
In the Borel hierarchy, Felix Hausdorff and Kazimierz Kuratowski proved that the countable levels of the
difference hierarchy over Π0γ give
Δ0γ+1.
References
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