surreal number

英语编辑

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surreal number

1974年由高德纳在其小说 Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness 中提出。此概念后来由英国数学家约翰·何顿·康威应用在对围棋博弈论的研究中。最初康威只称其为 numbers，但在1976年的 On Numbers and Games 中采用了高德纳的用词。

名词编辑

surreal number (复数 surreal numbers)

(数学) 超现实数
Conway's construction of surreal numbers relies on the use of transfinite induction.
康威对超现实数的建构基于超限归纳法的使用。

Conway's approach was to build numbers from scratch using a construction inspired by his game theory research; the resulting class of surreal numbers proved much larger than the class of real numbers.
（请为本使用例添加中文翻译）
- 1986, Harry Gonshor, An Introduction to the Theory of Surreal Numbers, Cambridge University Press, 1987, Paperback, ISBN 9780521312059.
- 2012, Fredrik Nordvall Forsberg, Anton Setzer, A Finite Axiomatisation of Inductive-Inductive Definitions, Ulrich Berger, Hannes Diener, Peter Schuster, Monika Seisenberger (editors), Logic, Construction, Computation, Ontos Verlag, page 263,
  The class² of surreal numbers is defined inductively, together with an order relation on surreal numbers wich is also defined inductively:
  • A surreal number $X=(X_{L},X_{R})$ consists of two sets $X_{L}$ and $X_{R}$ of surreal numbers, such that no element from $X_{L}$ is greater than any element from $X_{R}$ .
  
  • A surreal number $Y=(Y_{L},Y_{R})$ is greater than another surreal number $X=(X_{L},X_{R})$ , $X\leq Y$ , if and only if
  − there is no $x\in X_{L}$ such that $Y\leq x$ , and
  
  − there is no $y\in Y_{R}$ such that $y\leq X$ .
- 2018, Steven G. Krantz, Essentials of Mathematical Thinking, Taylor & Francis (Chapman & Hall/CRC Press), page 247,
  Here we shall follow Conway's exposition rather closely. Let $L$ and $R$ be two sets of numbers. Assume that no member of $L$ is greater than or equal to any member of $R$ . Then $\{L\vert R\}$ is a surreal number. All surreal numbers are constructed in this fashion.