Special Sets of Real Numbers

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9971282Bartoszynski In this project the investigators continue their research on variousset 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 linebetween ``large'' and ``small.'' In 1878 Cantor showed that the situationis far more complex, and the development of topology or analysis complicatedit even further. Today there are many different ways to express that a setof real numbers is small. Using the technique first developed by P.J. Cohen,one sees that many questions involving these smallness properties cannot beresolved in the framework of elementary mathematics. At the same time thestudy of these properties illuminated several problems from other areas ofmathematics, which gave this subject a life of its own.***