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.^[1]

In the Borel hierarchy, Felix Hausdorff and Kazimierz Kuratowski proved that the countable levels of the difference hierarchy over Π⁰_γ give Δ⁰_γ+1.^[2]

References

^ 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.
^ 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., vol. 37, Assoc. Symbol. Logic, La Jolla, CA, pp. 166–195, MR 2906999. See in particular p. 173.

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[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., vol. 37, Assoc. Symbol. Logic, La Jolla, CA, pp. 166–195, MR 2906999. See in particular p. 173.

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