Special Sets of Real Numbers

9971282 Bartoszynski In this project the investigators continue their research on various set theoretic notions of smallness in low levels of the cumulative hierarchy, particularly those that involve measure-theoretic, topological, Ramsey- theoretic, game-theoretic, or other combinatorial notions. Once ``infinite'' vs. ``finite'' was the mathematical dividing line between ``large'' and ``small.'' In 1878 Cantor showed that the situation is far more complex, and the development of topology or analysis complicated it even further. Today there are many different ways to express that a set of real numbers is small. Using the technique first developed by P.J. Cohen, one sees that many questions involving these smallness properties cannot be resolved in the framework of elementary mathematics. At the same time the study of these properties illuminated several problems from other areas of mathematics, which gave this subject a life of its own. ***