Difference hierarchy

In <a href="/facts/Set_theory/1ptfsvUP">set theory</a>, a branch of mathematics, the difference hierarchy over a <a href="/facts/Pointclass/k6fMNJa7">pointclass</a> is a <a href="/facts/Hierarchy_(mathematics)/bpcv4UuB">hierarchy</a> of larger pointclasses
generated by taking <a href="/facts/Complement_(set_theory)/yWUePIYY">differences</a> 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 <a href="/facts/Transfinite_number/m2oTjXaj">transfinite</a> to α-Γ for some <a href="/facts/Ordinal_number/GelFLjJt">ordinal</a> α.
In the <a href="/facts/Borel_sets/jvSsfpkP">Borel</a> hierarchy, <a href="/facts/Felix_Hausdorff/Rn6DuY2x">Felix Hausdorff</a> and <a href="/facts/Kazimierz_Kuratowski/8xoKvo5b">Kazimierz Kuratowski</a> proved that the <a href="/facts/Countable/Wj4DmWPv">countable</a> levels of the 
difference hierarchy over Π0γ give
Δ0γ+1.

Difference hierarchy open-in-new

Difference hierarchy