超实数上的积分
Integration on the Surreals
AI总结：
本文证明实际应用中的多数函数，尤其是复苏函数的一大子类，可构造性扩展到超实数域并定义积分，同时指出更一般函数的扩展受基础数学限制。
摘要：
Conway 的超实数实闭域 $\mathbf{No}$ 是实数与序数的一种广泛推广，已有研究表明，log 和指数等许多初等函数可以扩展到该域。然而，识别能够如此扩展的重要函数类别，并为它们定义积分，这些问题一直极为困难。本文通过证明实际应用中出现的大多数函数都存在到 $\mathbf{No}$ 的扩展，从而也存在积分，来解决这一问题及相关未决议题。特别地，我们证明这些扩展和积分对 resurgent functions（复苏函数）的一个很大子类存在；该子类包含在 $\infty$ 处为半代数、半解析、解析、亚纯以及 Borel 可和的函数，也包含非共振线性及非线性亚纯 ODE 系统或差分方程系统的解。通过适当的变量替换，我们可以处理任意位置的奇点。我们进一步建立了一个充分条件，使该理论能够更一般地迁移到 $\mathbf{No}$ 的有序指数子域，并借助超实数文献中常见的结构来说明所得结果。我们所关注的函数扩展与积分在本质上是构造性的，这使我们能够在不含选择公理（同时针对集合和真类）的 NBG 中工作。在论文的肯定性部分完成之后，本文还表明：由于数学基础方面的考量，为实质上更一般的函数类型（例如光滑函数）构造此类构造性扩展和积分，其存在性会受到阻碍。
英文摘要：
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.
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