论文标题
超现实的整合
Integration on the Surreals
论文作者
论文摘要
Conway的真实封闭字段$ \ Mathbf {no}超现实数字的$是对实数的广泛概括，并且显示了许多基本功能（例如日志和启动）的序数。证明可以扩展的重要函数类别并为其定义集成的问题被证明是可怕的。在本文中，我们通过表明扩展到$ \ mathbf {no} $的扩展，从而解决此问题和相关的未解决的问题，从而在实际应用中出现的大多数函数都存在积分。 In particular, we show they exist for a large subclass of the \emph{resurgent functions}, a subclass that contains the functions that at $\infty$ are semi-algebraic, semi-analytic, analytic, meromorphic, and Borel summable as well as solutions to nonresonant linear and nonlinear meromorphic systems of ODEs or of difference equations.通过适当的变量更改，我们处理任意定义的奇异点。我们进一步建立了足够的条件，可以使该理论更加普遍地将$ \ mathbf {no} $的指数级子领域延续到有序的指数子场中，并用超现实文献中熟悉的结构说明了结果。关注我们的函数和积分的扩展本质上是建设性的，这使我们能够在NBG中工作较少选择的公理（对于集合和适当的类）。在本文的正面部分完成后，这表明存在这种建设性扩展的存在和基本类型的功能（例如平滑函数）的积分被数学基础所考虑的因素所阻碍。
Conway's real closed field $\mathbf{No}$ of surreal numbers is a sweeping generalization of the real numbers and the ordinals to which a number of elementary functions such as log and exponentiation have been shown to extend. The problems of identifying significant classes of functions that can be so extended and of defining integration for them have proven to be formidable. In this paper we address this and related unresolved issues by showing that extensions to $\mathbf{No}$, and thereby integrals, exist for most functions arising in practical applications. In particular, we show they exist for a large subclass of the \emph{resurgent functions}, a subclass that contains the functions that at $\infty$ are semi-algebraic, semi-analytic, analytic, meromorphic, and Borel summable as well as solutions to nonresonant linear and nonlinear meromorphic systems of ODEs or of difference equations. By suitable changes of variables we deal with arbitrarily located singular points. We further establish a sufficient condition for the theory to carry over to ordered exponential subfields of $\mathbf{No}$ more generally and illustrate the result with structures familiar from the surreal literature. The extensions of functions and integrals that concern us are constructive in nature, which permits us to work in NBG less the Axiom of Choice (for both sets and proper classes). Following the completion of the positive portion of the paper, it is shown that the existence of such constructive extensions and integrals of substantially more general types of functions (e.g. smooth functions) is obstructed by considerations from the foundations of mathematics.
