surreal number

英語编辑

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surreal number

1974年由高德納在其小說 Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness 中提出。此概念後來由英國數學家約翰·何頓·康威應用在對圍棋博弈論的研究中。最初康威只稱其為 numbers，但在1976年的 On Numbers and Games 中採用了高德納的用詞。

名詞编辑

surreal number (複數 surreal numbers)

(數學) 超現實數
Conway's construction of surreal numbers relies on the use of transfinite induction.
康威對超現實數的建構基於超限歸納法的使用。

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.
（請為本使用例添加中文翻譯）
- 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.