Difference hierarchy

From Wikipedia, the free encyclopedia

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.[1]

In the Borel hierarchy, Felix Hausdorff and Kazimierz Kuratowski proved that the countable levels of the difference hierarchy over Π0γ give Δ0γ+1.[2]

References

  1. ^ Kanamori, Akihiro (2009), The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings, Springer Monographs in Mathematics (2nd ed.), Springer-Verlag, Berlin, p. 442, ISBN 978-3-540-88866-6, MR 2731169 .
  2. ^ Wadge, William W. (2012), "Early investigations of the degrees of Borel sets", Wadge degrees and projective ordinals. The Cabal Seminar. Volume II, Lect. Notes Log., 37, Assoc. Symbol. Logic, La Jolla, CA, pp. 166–195, MR 2906999 . See in particular p. 173.


Retrieved from "https://en.wikipedia.org/w/index.php?title=Difference_hierarchy&oldid=688279888"
This content was retrieved from Wikipedia : http://en.wikipedia.org/wiki/Difference_hierarchy
This page is based on the copyrighted Wikipedia article "Difference hierarchy"; it is used under the Creative Commons Attribution-ShareAlike 3.0 Unported License (CC-BY-SA). You may redistribute it, verbatim or modified, providing that you comply with the terms of the CC-BY-SA