Applications of Set Theory to Analysis

自从20世纪30年代库尔特·戈德尔的工作以来，人们就知道，数学中有一些既不能证明也不能反驳的陈述；但这些陈述完全是人为的，可以说它们永远不会出现在任何自然环境中。然而，保罗·科恩在20世纪60年代初表明，有些陈述无法确定哪些数学家真正关心，哪些陈述自然地出现在他们的学科中。自那以后，他为此开发的技术已经得到了许多应用，现在无法决定的陈述来自人们可能猜不到的领域，可以找到不可决定的东西。这项提议延续了在数学中寻找陈述的计划，准确地说，在这些现象发生的地方寻找分析。 这个项目对加拿大科学的意义在于，它为培养加拿大下一代研究数学家提供了一种工具。为了使加拿大受益于全球科学进步，特别是数学进步，它必须支持一批国际公认对全球研究事业作出重大贡献的研究人员。该研究项目将允许有才华的学生和博士后研究员建立声誉，使他们能够参与加拿大科学可以受益的国际尖端研究项目。

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.