Ideals Of Real Numbers In Descriptive Set Theory (measure, Meager, Games).
Robert Anthony Mayans · Deep Blue (University of Michigan) · 1985
The ideals of Borel sets on the unit interval, closed under countable unions and invariant under translations, include two important examples: the sets of Lebesgue measure zero, and the sets of first Baire category (or meager sets). Using a stronger notion of translation, called a rearrangement, one can classify the ideals closed under rearrangements and countable unions. They are the two ideals mentioned, their intersection, and the ideal of countable sets. The proof requires only the standard regularity properties of Borel sets and may be extended, for example, to ideals of projective sets if Projective Determinacy is assumed, or to ideals of arbitrary sets in the Levy-Solovay model. The other topic of the dissertation is the ideals of real numbers associated with certain infinite games, including the Banach-Mazur game. I start with a definition of games of closed sets of real numbers, and concentrate on the case when the collection of winning sets for the second player is closed under countable unions. If an ideal satisfies a simple approximation condition for a closed set game, then the ideal characterizes the sets for which the second player has a winning strategy, and the game is determined for analytic sets (in ordinary ZFC). In addition, a forcing argument shows that for games of this type, all sets are determined in the Levy-Solovay model. Some new restrictions are needed to make the games and the strategies portable between models. This method gives several familiar regularity properties for sets of reals in the Levy-Solovay model, as well as some new ones, in a uniform game-theoretic proof.