title: Difference hierarchy
text: # 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 $\{A:\exists C,D\in \Gamma (A=C\setminus D)\}$. 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.
  
  In the Borel hierarchy, Felix Hausdorff and Kazimierz Kuratowski proved that the countable levels of the difference hierarchy over $Π^0_γ$ give $Δ^0_{γ+1}.$
  
  