Wadge hierarchy

In descriptive set theory, within mathematics, Wadge degrees are levels of complexity for sets of reals. Sets are compared by continuous reductions. The Wadge hierarchy is the structure of Wadge degrees. These concepts are named after William W. Wadge. Suppose and are subsets of Baire space ωω. Then is Wadge reducible to or ≤W if there is a continuous function on ωω with . The Wadge order is the preorder or quasiorder on the subsets of Baire space. 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. 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 hierarchy and the difference hierarchy. The Wadge hierarchy plays 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 algorithmic complexity. Wadge's lemma states that under the axiom of determinacy (AD), for any two subsets of Baire space, ≤W or ≤W ωω. The assertion that the Wadge lemma holds for sets in Γ is the semilinear ordering principle for Γ or SLO(Γ). Any defines a linear order on the equivalence classes modulo complements. Wadge's lemma can be applied locally to any pointclass Γ, for example the Borel sets, Δ1n sets, Σ1n sets, or Π1n sets. It follows from determinacy of differences of sets in Γ. Since Borel determinacy is proved in ZFC, ZFC implies Wadge's lemma for Borel sets. Wadge's lemma is similar to the cone lemma from computability theory. 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.