论文标题
在指数，对数，衍生和抗衍生功能下稳定的超现实场稳定
Surreal fields stable under exponential, logarithmic, derivative and anti-derivative functions
论文作者
论文摘要
最初由Conway提出的$ \ textbf {no} $表示的超现实数字类是一个通用有序字段，因为任何有序字段都可以嵌入其中。它们特别包括实数和序数。他们与其他领域（例如跨系列领域）有着密切的关系。在贡献之后，超现实数字可以看作是序数长度的符号序列，具有一些指数和对数函数，这些函数将通常的函数扩展到真实的函数上。 $ \ textbf {no} $实际上可以看作是一个具有真实系数的优雅（广义）功率系列字段，即Hahn系列，带有$ \ textbf {no} $本身的指数。 几年前，Berarducci和Mantova认为超现实数字的衍生是与跨性别相通的功能的衍生作用。在本文中，在我们先前的工作之后，我们在指数，对数，派生和抗衍生的所有操作下对超现实场的结构表现出足够的条件。从长远来看，通过可计算性考虑，我们还提供了该定理的非平凡应用：存在一个相当合理的领域，该领域仅需要高达$ε_Ω$，该领域远小于$ω_1^{ck} $（resp。$ω_1$），第一次是不可接受的（nontrable-compable forsical）。
The class of surreal numbers, denoted by $\textbf{No}$, initially proposed by Conway, is a universal ordered field in the sense that any ordered field can be embedded in it. They include in particular the real numbers and the ordinal numbers. They have strong relations with other fields such as field of transseries. Following Gonshor, surreal numbers can be seen as signs sequences of ordinal length, with some exponential and logarithmic functions that extend the usual functions over the reals. $\textbf{No}$ can actually be seen as an elegant (generalized) power series field with real coefficients, namely Hahn series with exponents in $\textbf{No}$ itself. Some years ago, Berarducci and Mantova considered derivation over the surreal numbers, seeing them as germs of functions, in correspondence to transseries. In this article, following our previous work, we exhibit a sufficient condition on the structure of a surreal field to be stable under all operations among exponential, logarithm, derivation and anti-derivation. Motivated, in the long term, by computability considerations, we also provide a non-trivial application of this theorem: the existence of a pretty reasonable field that only requires ordinals up to $ε_ω$, which is far smaller than $ω_1^{CK}$ (resp. $ω_1$), the first non-computable (resp. uncountable) ordinal.
