Difference hierarchy explained

In set theory, a branch of mathematics, the difference hierarchy over a pointclass is a hierarchy of larger pointclassesgenerated by taking differences of sets. If Γ is a pointclass, then the set of differences in Γ is

\{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 : \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.^[1]

In the Borel hierarchy, Felix Hausdorff and Kazimierz Kuratowski proved that the countable levels of the difference hierarchy over Π⁰_{γ giveΔ⁰γ+1.^[2]}

Notes and References

1. .
2. . See in particular p. 173.