Cardinal Invariants and Sets of Reals

这些研究项目涉及实数线的集合论，是描述集合论的一部分。主要研究者研究实数集合的各种理想。连续统的基本特征和相关的小集合家族是本研究的核心。具体地说，他着重于关于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合族完全由可数集合组成的陈述。他还研究了两个经典的小概念：度量零和第一类之间的对偶程度。这个问题涉及找到度量概念之间的平行关系，如强度量零和通用度量零，以及它们的第一类类似物。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。测度和范畴的概念经过了大约一百年的严格研究，并成功地应用于现代数学的许多领域。Bartoszynski在这一领域探讨了几个问题。这些问题可能产生作为数学定理的正答案，或者它们可能独立于集合论的标准公理。积极的答案涉及有限和无限组合学的新结果，并且已经超越了传统的集合理论，应用到现实分析，测量理论和拓扑。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.