Mathematical Sciences: The Set Theory of the Real Line

在集合论中的各种问题的真实的线将是 研究了 这些问题涉及独立性， 一致性导致公理集合论。 这意味着 问题是要表明某些特定的陈述不能被 由集合论的一般公理证明。 使用的主要技术是强迫。 强迫的方法是 第一次发明的保罗J科恩在六十年代，以表明， 连续统假设独立于通常的集合公理 理论 从那以后，它被用来显示许多问题，不仅是 集合论，还有代数、拓扑和分析， 独立于集合论的一般公理。 两个领域的集合论的真实的线将是 研究了 第一个领域考虑勒贝格性质 测度和Baire范畴在真实的直线上以及 不太为人所知的理想。 第二个领域是 研究真实的线的病理或特殊子集。 理解真实的直线的集合论性质 明显的基础性意义。 还透露， 为此目的开发的方法启发了类似物， 在理论计算机科学的背景下。

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.