Cardinal Invariants and Sets of Reals

这些研究项目涉及实线集合论，这是描述集合论的一部分。主要研究人员研究实数集的各种理想。连续统和相关的小集族的基数特征是这项研究的中心。具体地说，他专注于Borel猜想的各种推广的一致性问题，Borel猜想是一个断言所讨论的集族完全由可数集组成的陈述。他还研究了两个经典的小概念之间的对偶程度：测量零和第一个范畴。实数结构的研究标志着集合论的起源，自上世纪初以来一直是系统研究的对象。度量和范畴的概念已经被严格地研究了大约一百年，并被成功地应用于现代数学的许多领域。Bartoszynski在这一领域探索了几个问题。这些问题可能会产生数学定理的肯定答案，或者它们可能被证明独立于集合论的标准公理。肯定的答案涉及有限和无限组合学的新结果，并具有超越传统集合论到重新分析、测度论和拓扑学的应用。另一方面，独立性结果，特别是那些使用强迫的结果，在集合论本身中也有应用。

These research projects concern the set theory of the real line,a part of descriptive set theory. The principal investigatorstudies various ideals of sets of real numbers. Cardinalcharacteristics of the continuum and the associated families ofsmall sets are central to this research. Specifically, hefocuses on problems concerning the consistency of variousgeneralizations of the Borel Conjecture, a statement assertingthat the families of sets in question consist entirely ofcountable sets. He also studies the extent of the dualitybetween the two classical notions of smallness: measure zero andthe first category. This problem concerns finding parallelsbetween the measure concepts such as strong measure zero anduniversal measure zero, and their first category analogs.The study of the structure of the real numbers marks the originsof set theory and has been the object of systematic researchsince the beginning of the last century. Concepts of measure andcategory have been studied rigorously for about one hundredyears, and have been successfully used in many areas of modernmathematics. Bartoszynski pursues several problems in this area.These problems may yield positive answers that are theorems ofmathematics or they may turn out to be independent from thestandard axioms of set theory. The positive answers involve newresults in both finite and infinite combinatorics and haveapplications reaching beyond traditional set theory to realanalysis, measure theory, and topology. On the other hand, theindependence results, particularly ones using forcing, haveapplications within set theory itself.