Difference hierarchy

In set theory, 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 ) } . 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 ) ) } . 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 Π⁰_γ give Δ⁰_γ+1.