Mathematical Sciences: The Set Theory of the Real Line

实线集合论中的各种问题将被研究。这些问题涉及公理化集合论中的独立性和一致性结果。这意味着问题是要证明某些特定的命题不能从集合论的通常公理中得到证明。使用的主要技术是强迫。强迫法最初是由保罗·j·科恩在60年代发明的，目的是证明连续统假设独立于集合论的常用公理。它已被用来证明许多问题，不仅是集合论，而且代数，拓扑和分析是独立于集合论的通常公理。实线集合论的两个领域将被研究。第一部分考虑了勒贝格测度和贝尔范畴在实线上的性质，以及一些不太为人所知的理想。第二个领域是调查真实线的病理或特殊子集。认识实线的集论性质具有明显的基础意义。它还揭示了为此目的开发的方法激发了在理论计算机科学背景下工作的类似物。

Various problems in the set theory of the real line will be investigated. These problems involve independence and consistency results in axiomatic set theory. This means that the problem is to show that some particular statement cannot be proved from the usual axioms of set theory. The main technique used is forcing. The forcing method was first invented by Paul J. Cohen in the sixties to show that the continuum hypothesis is independent of the usual axioms of set theory. It has since been used to show many problems not only of set theory, but also algebra, topology, and analysis are independent of the usual axioms of set theory. Two areas of the set theory of the real line will be investigated. The first area considers properties of Lebesgue measure and Baire category on the real line as well as some ideals which are less well known. The second area is to investigate pathological or peculiar subsets of the real line. Understanding set theoretic properties of the real line has obvious foundational significance. It has also transpired that methods developed for this purpose inspire analogues which work in the context of theoretical computer science.