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 if and only if it is in the second ambiguous class (), if and only if 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 , a guessable set is annihilated by applications of the simplified remainder if and only if it is guessable with fewer than mind changes. We use guessability with fewer than 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 for all successor ordinals .
"Guessing, Mind-Changing, and the Second Ambiguous Class." Notre Dame J. Formal Logic 57 (2) 209 - 220, 2016. https://doi.org/10.1215/00294527-3443549