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 $\backslash \{A:\backslash exists\; C,D\backslash in\backslash Gamma\; (\; A\; =\; C\backslash setminus\; D)\backslash \}$. 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:

$\backslash \{A\; :\; \backslash exists\; C,D,E\backslash in\backslash Gamma\; (\; A=C\backslash setminus(D\backslash setminus\; E))\backslash \}$. This definition can be extended recursively into the transfinite to α-Γ for some ordinal α.{{citation

| last = Kanamori | first = Akihiro

| authorlink = Akihiro Kanamori

| edition = 2nd

| isbn = 978-3-540-88866-6

| mr = 2731169

| at = [https://books.google.com/books?id=3TQ_AAAAQBAJ&pg=PA442 p. 442]

| publisher = Springer-Verlag, Berlin

| series = Springer Monographs in Mathematics

| title = The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings

| title-link = The Higher Infinite

| year = 2009}}.

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.{{citation

| last = Wadge | first = William W.

| contribution = Early investigations of the degrees of Borel sets

| mr = 2906999

| pages = 166–195

| publisher = Assoc. Symbol. Logic, La Jolla, CA

| series = Lect. Notes Log.

| title = Wadge degrees and projective ordinals. The Cabal Seminar. Volume II

| volume = 37

| year = 2012}}. See in particular [https://books.google.com/books?id=ys1uokzTdX0C&pg=PA173 p. 173].