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I have been researching Combinatorial Game Theory. One common theme is the assignment of values to games in order to classify the game as a win for a specific player. One such way is class of surreal numbers which is essentially defined as an ordered pair of surreal numbers.

However, the surreal numbers are mostly useful for analyzing partizan games. For impartial games, the values are fuzzy surreal numbers. Consequentially, in the analysis of impartial games, the concept of nimbers and Sprague-Grundy Values are used. Nimbers essentially relate a position in a normal impartial game to a nim heap.

My question is that can the nimbers be interpreted in the concept of surreal numbers. For example, the surreal number $* = \{0|0\}$ corresponds to a Sprague-Grundy Value of $1$. Can this approach be extended to all Sprague-Grundy values in such a way that the bitwise xor definition of nimber addition coincides the surreal number addition?

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    $\begingroup$ Both surreal numbers and nimbers are part of a more general Abelian group of games, but there is certainly no surreal number corresponding so directly to any nonzero nimber; nonzero nimbers have order 2 and nonzero surreal numbers have infinite order. $\endgroup$
    – Gabriel C. Drummond-Cole
    Commented Jul 17, 2015 at 5:09
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    $\begingroup$ The short answer to the question is no. By the way, the game * = {0|0} is not a number, and corresponds to the Sprague-Grundy value 1, not 0. $\endgroup$
    – Johan Wästlund
    Commented Jul 17, 2015 at 10:01
  • $\begingroup$ Sorry that was a typo. The game $\{|\}$ corresponds to $0$. $\endgroup$
    – Halbort
    Commented Jul 17, 2015 at 16:58
  • $\begingroup$ Be nice to move this question to the other Maths site if it better belongs there. $\endgroup$
    – alan2here
    Commented Sep 2, 2016 at 15:49

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