论文标题
在指数和对数功能下稳定的超现实场
Surreal fields stable under exponential and logarithmic functions
论文作者
论文摘要
超现实的数字，具有非常丰富而优雅的理论。这类数字（用NO表示）同时包括序数和实数，并形成了一个通用的巨大实际封闭字段：从某种意义上说，任何真实的封闭字段都可以嵌入其中。在贡献之后，超现实数字也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函数上。实际上，No也可以看作是一个具有真实系数的优雅（广义）功率系列领域，即Hahn系列具有No本身的指数。它也可以被视为特定的跨系列领域，为连续体的功能提供了一些分析和渐近分析的工具，提供了讨论过度族裔或sublogarithm函数的自然概念及其渐近学。在本文中，我们考虑了在指数和对数函数下NO的子场的稳定性。也就是说，我们考虑了集的超现实数字，其符号序列的长度小于某些序数λ。扩展了Van Den Dries和Ehrlich的讨论，我们表明，通过指数和对数Iffλ的稳定性是某种εnumber。长期通过使用序数机器的计算性问题，我们通过Hahn系列定义的指数和对数函数来考虑子场Stable，而这不需要提高基本长度和指数。我们证明，不能通过指数和对数函数来表示不可表达的子字段的严格层次结构。这提供了许多通过指数和对数函数稳定的子字段的明确示例，并且不需要登上红衣主教λ来提供此类示例。
Surreal numbers, have a very rich and elegant theory. This class of numbers, denoted by No, includes simultaneously the ordinal numbers and the real numbers, and forms a universal huge real closed field: It is universal in the sense that any real closed field can be embedded in it. Following Gonshor, surreal numbers can also be seen as signs sequences of ordinal length, with some exponential and logarithmic functions that extend the usual functions over the reals. No can actually also be seen as an elegant particular (generalized) power series field with real coefficients, namely Hahn series with exponents in No itself. It can also be considered as a particular field of transseries, providing tools to do some analysis and asymptotic analysis for functions over the continuum, providing natural concepts for discussing hyperexponential or sublogarithm functions, and their asymptotics. In this article, we consider stability of subfields of No under exponential and logarithmic functions. Namely, we consider the set surreal numbers whose signs sequences have length less than some ordinal λ. Extending the discussion from van den Dries and Ehrlich, we show that is stable by exponential and logarithm iff λ is some ε-number. Motivated in a longer term by computability issues using ordinal machines, we consider subfields stables by exponential and logarithmic functions defined by Hahn series that does not require to go up to cardinal lengths and exponents. We prove that No can be expressed as a strict hierarchy of subfields stable by exponential and logarithmic functions. This provides many explicit examples of subfields of No stable by exponential and logarithmic functions, and does not require to go up to a cardinal λ to provide such examples.
