Structure of The Wadge Hierarchy
Martin and Monk proved in 1973 that AD implies the Wadge order for Baire space is well founded. 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).
As for the Wadge lemma, this holds for any pointclass Γ, assuming the axiom of 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 α≤θ the collection Wα of sets which show up before stage α is a pointclass. Conversely, every pointclass is equal to some α. A pointclass is said to be self-dual if it is closed under complementation. It can be shown that Wα is self-dual if and only if α is either 0, an even successor ordinal, or a limit ordinal of countable cofinality.
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