Special Sets of Real Numbers

9971282Bartoszynski在这个项目中，研究人员继续他们对累积层次低水平上的小的各种集合论概念的研究，特别是那些涉及测度论、拓扑学、拉姆齐理论、博弈论或其他组合概念的概念。曾经，“无限”和“有限”是“大”和“小”之间的数学分界线。1878年，康托指出，情况要复杂得多，拓扑学或分析的发展使情况变得更加复杂。今天，有许多不同的方式来表示一组实数是小的。使用P.J.科恩最先开发的技术，人们可以看到，许多涉及这些微小性质的问题不能在初等数学的框架内解决。与此同时，对这些性质的研究启发了其他数学领域的几个问题，这些问题赋予了这门学科自己的生命。

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.***