- Wadge hierarchy
descriptive set theory, Wadge degrees are levels of complexity for sets of reals and more comprehensively, subsets of any given topological space. Sets are compared by continuousreductions. The Wadge hierarchy is the structure of Wadge degrees.
Take a subset of and a subset of , where and are topological spaces. is Wadge reducible to , denoted by : if and only if there is a continuous function from to such that . The notion of reduction depends on the spaces and .
For a given topological space , Wadge reducibility determines a
preorderor quasiorderon the subsets of . This is the Wadge order for . Equivalence classes of sets under this preorder are called Wadge degrees, the degree of a set is denoted by  W. The set of Wadge degrees ordered by the Wadge order is called the Wadge hierarchy. Wadge degrees are usually considered for Baire space ωω, Cantor space, and other Polish spaces. If the underlying space is not specified, the terms "Wadge degree" and "Wadge hierarchy" usually refer to Baire space.
Properties of Wadge degrees include their consistency with measures of complexity stated in terms of definability. For example, if ≤W and is a countable intersection of
open sets, then so is . The same works for all levels of the Borel hierarchyand the difference hierarchy.
Wadge degrees for Baire space play an important role in models of the
axiom of determinacy.
Further interest in Wadge degrees comes from computer science, where some papers have suggested Wadge degrees are relevant to
The Wadge game is a simple infinite game discovered by William Wadge (pronounced "wage"). It is used to investigate the notion of continuous reduction for subsets of Baire space. Wadge had analyzed the structure of the Wadge hierarchy for Baire space with games by 1972, but published these results only much later in his PhD thesis. In the Wadge game , player I and player II each in turn play integers which may depend on those played before. The outcome of the game is determined by checking whether the sequences x and y generated by players I and II are contained in the sets A and B, respectively. Player II wins if the outcome is the same for both players, i.e.: Player I wins if the outcome is different. Sometimes this is also called the "Lipschitz game", and the variant where player II has the option to pass (but has to play infinitely often) is called the Wadge game.
Suppose for a moment that the game is determined. If player I has a winning strategy, then this defines a continuous (even Lipschitz) map reducing to the complement of , and if on the other hand player II has a winning strategy then you have a reduction of to . For example, suppose that player II has a winning strategy. Map every sequence x to the sequence y that player II plays in if player I plays the sequence x, where player II follows his or her winning strategy. This defines a is a continuous map f with the property that x is in if and only if f(x) is in .
Wadge's lemma states that under the
axiom of determinacy(AD), for any two subsets of Baire space: This works since one of the players has a winning strategy in the Wadge game , and this strategy defines one of the continuous maps needed. The lemma can also be applied locally to pointclasses Γ, for example the Borel sets, Δ1n sets, Σ1n sets, or Π1n sets. Here the conclusion of Wadge's lemma for sets in Γ follows from determinacy of Booleancombinations of sets in Γ instead of full AD. Since Borel determinacy is proved in ZFC, ZFC implies Wadge's lemma for Borel sets.
The assertion that the Wadge lemma holds for sets in Γ is the "semilinear ordering principle" for Γ, SLO(Γ). Any semilinear order defines a linear order on the equivalence classes modulo complements. In this case the relation on the set of pairs ωω given by ωω ≤W ωω if and only if ≤W or ≤W ωω, is linear.
Hence under AD, the Wadge classes modulo complements form a wellorder. The Wadge rank of a set is the order type of the set of Wadge degrees modulo complements strictly below  W.
The length of the Wadge hierarchy has been shown to be Θ. Wadge also proved that the length of the Wadge hierarchy restricted to the Borel sets is φω1(1) (or φω1(2) depending on the notation), where φγ is the γth Veblen function to the base ω1 (instead of the usual ω).
Structure of the Wadge hierarchy
Continuing to work under determinacy, if we associate with each set the collection of all sets strictly below on the Wadge hierarchy, this forms a pointclass. Equivalently, for each ordinal
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