Replica trick: Difference between revisions
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{{Noref|date=May 2011}} | |||
In [[set theory]], the '''difference hierarchy''' over a [[pointclass]] is a [[hierarchy (mathematics)|hierarchy]] of larger pointclasses | |||
generated by taking [[complement (set theory)|difference]]s of sets. If Γ is a pointclass, then the set of differences in Γ is <math>\{A:\exists C,D\in\Gamma ( A = C\setminus D)\}</math>. 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: | |||
<math>\{A : \exists C,D,E\in\Gamma ( A=C\setminus(D\setminus E))\}</math>. This definition can be extended recursively into the transfinite to α-Γ for some [[ordinal number|ordinal]] α. | |||
In the [[Borel sets|Borel]] and [[projective set|projective hierarchies]], [[Felix Hausdorff]] proved that the countable levels of the | |||
difference hierarchy over Π<sup>0</sup><sub style="margin-left:-0.6em">γ</sub> and Π<sup>1</sup><sub style="margin-left:-0.6em">γ</sub> give | |||
Δ<sup>0</sup><sub style="margin-left:-0.6em">γ+1</sub> and Δ<sup>1</sup><sub style="margin-left:-0.6em">γ+1</sub>, respectively. | |||
{{settheory-stub}} | |||
[[Category:Descriptive set theory]] | |||
[[Category:Mathematical logic hierarchies]] | |||
Revision as of 06:58, 29 January 2014
Template:Noref 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 . 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: . This definition can be extended recursively into the transfinite to α-Γ for some ordinal α.
In the Borel and projective hierarchies, Felix Hausdorff proved that the countable levels of the difference hierarchy over Π0γ and Π1γ give Δ0γ+1 and Δ1γ+1, respectively.