Guessing, Mind-Changing, and the Second Ambiguous Class

In his dissertation, Wadge defined a notion of guessability on subsets of the Baire space and gave two characterizations of guessable sets. A set is guessable iff it is in the second ambiguous class (∆2), iff it is eventually annihilated by a certain remainder. We simplify this remainder and give a new proof of the latter equivalence. We then introduce a notion of guessing with an ordinal limit on how often one can change one’s mind. We show that for every ordinal α , a guessable set is annihilated by α applications of the simplified remainder if and only if it is guessable with fewer than α mind changes. We use guessability with fewer than α mind changes to give a semi-characterization of the Hausdorff difference hierarchy, and indicate how Wadge’s notion of guessability can be generalized to higher-order guessability, providing characterizations of ∆α for all successor ordinals α > 1.