1 Answers
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 : ∃ C , D ∈ Γ } {\displaystyle \{A:\exists C,D\in \Gamma \}}. 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 ] } {\displaystyle \{A:\exists C,D,E\in \Gamma ]\}}. 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 Πγ giveΔγ+1.