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.

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