Wadge hierarchy and Veblen hierarchy Part I: Borel sets of finite rank

Abstract

We consider Borel sets of the form A ! (with usual topology) where cardinality of is less than some uncountable regular cardinal . We obtain a \normal form" of A, by nding a Borel set ( ) such that A and ( ) continuously reduce to each other. We do so by de ning Borel operations which are homomorphic to the rst Veblen ordinal functions of base required to compute the Wadge degree of the set A: the ordinal .I starts and plays letters in 33 II plays letters in T A ( T denotes the alphabet of the Tree T and A is the set of arrows of di erent type involved by operation )

Game Characterizations and Lower Cones in the Weihrauch Degrees

“…For example, in the eraser game (implicit in [Dup01]) characterizing the Baire class 1 functions, player II is allowed to erase past moves, the rule being that she is only allowed to erase each position of her output sequence finitely often. In the backtrack game (implicit in [Van78]) characterizing the functions which preserve the class of Σ 0 2 sets under preimages, player II is allowed to erase all of her past moves at any given round, the rule in this case being that she only do this finitely many times.…”

Transfinite Extension of the Mu-Calculus

“…Thus, α is of the form α = (ω 1 ) δ+1 . Sets of such a Wadge rank have a very nice representation, as it was proved in [4]. We will use this representation in a crucial way.…”

“…The reason for this is the following: Our construction uses in a crucial way a structural theorem of [4]. The effective version is not proved.…”

“…Surprisingly, the progression is precisely multiplication by ω ck 1 . In the boldface hierarchy Duparc [4], extending Wadge's [15] definition on self-dual sets, defined an operation which increases the Wadge rank of a given set by the factor ω 1 . Namely,…”

“…Following [4], a conciliatory set is a set A ⊆ Σ * ∪ Σ ω . The Wadge hierarchy can be extended to conciliatory sets, by allowing I to skip as well as II.…”