Applications of Set Theory to Analysis

自从库尔特·哥德尔在20世纪30年代的工作以来，人们就知道数学中有一些陈述既不能被证明也不能被证伪;但是这些陈述是相当人为的，可以说它们永远不会出现在任何自然环境中。然而，在20世纪60年代初，保罗·科恩（Paul Cohen）指出，有些陈述无法确定哪些数学家真正关心，哪些陈述在他们的学科中自然出现。他开发的技术，这样做，因为发现了许多应用程序和不可判定的语句现在来自的领域，人们不会猜到不可判定性可以发现。这一建议继续程序寻找语句在数学，分析是准确的，在这种现象发生。 该项目对加拿大科学的重要性在于，它为培养下一代加拿大数学研究人员提供了一个工具。为了使加拿大从全球科学进步中受益，特别是数学，它必须支持一批在国际上被公认为对全球研究事业做出重大贡献的研究人员。这个研究项目将使有才华的学生和博士后研究员建立声誉，使他们能够参与国际，尖端的研究项目，加拿大科学可以从中受益。

It has been know since the work of Kurt Godel in the 1930s that there are statements in mathematics that can neither be proved nor disproved; but these statements were quite artificial and it could be argued that they would never arise in any natural context. However, in the early 1960s Paul Cohen showed that there are statements that can not be decided about which mathematicians actually care and which arise naturally in their subject. The technique he developed for doing this has since found many applications and the undecidable statements now come from areas where one would not have guessed undecidability could be found. This proposal continues the program of looking for statements in mathematics, analysis to be precise, where such phenomena occur. The significance of this project to Canadian science is that it provides a vehicle for training the next generation of Canada's research mathematicians. In order for Canada to benefit from global advances in science, mathematics in particular, it must support a body of researchers who are acknowledged internationally to be contributing to the global research enterprise in a significant way. This research project will allow talented students and postdoctoral fellows to establish reputations that will allow them to participate in international, cutting edge research projects from which Canadian science can benefit.