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Surreal numbers are the largest possible structure to have a complete order. Games are an extension of the Surreals which only admits a partial order. Along with being larger, smaller or equal to each other, two Games can be fuzzy. For instance, $\left(0|0\right)=*$ is fuzzy to $0$ - they are neither larger, nor smaller nor equal to each other.

Now I wonder: What is left of the Games if you take only those which are countable or maybe computable? Is there a nice construction that more or less directly gives you the countable subset of all Games in a finite number of steps? (Typical constructions I know of would take countably infinitely many steps to construct $\frac{1}{3}$ or any other fraction that isn't of the form $\frac{n}{2^k}$ with $n \in \mathbb{Z}$ and $k \in \mathbb{N}$.)

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  • $\begingroup$ What do you mean by the phrases "those which are countable", "computable game", or "the countable subset of all games in a finite number of steps"? If you have to output countably many things, and outputting one thing is a step, you're not going to be able to do it in finitely many steps. You could take the computable sign expansions (analogous to computable binary expansions of reals), but even if that's what you want, I'm not sure what question you're asking about them. $\endgroup$
    – Mark S.
    Sep 26 '14 at 1:02
  • $\begingroup$ @MarkS. all I want is a way to nicely express fractions / use division over a subset of the surreals. Obviously I could just use rational numbers. But I'd like to have access to combinatorial games too. The sign expansion, if I'm not mistaken, will still not give me easy access to, say, 1/3. That still requires an infinite expansion. Obviously, so does its representation as a real number. But I want a finite, exact representation of arbitrary fractions. On top of being infinite for non-power-of-2 fractions, the sign expansion can't represent non-surreal games, can it? $\endgroup$
    – kram1032
    Oct 5 '14 at 11:53
  • $\begingroup$ To address your last question first, division doesn't generally work out for non-surreal games, and to the extent that it does, it's not unique: Should * be considered 0/2 since *+*=0? You are likely doomed if you're looking to go beyond the surreals. -- The nice way to use division over some surreals with finite representations is to use the standard notation for rational numbers and ordinals, etc. $1/\omega+1/(1+1/\omega_1)$ is a finite string representing a surreal number that's not a rational. $1/3$ is a finite representationing a surreal number with infinite sign expansion. etc. $\endgroup$
    – Mark S.
    Oct 13 '14 at 2:45
  • $\begingroup$ If you want to go beyond the surreals and use division, you're doomed because division doesn't generally make sense (and to the extent that it would inherit sense from game addition, it's not well defined: *+*=0, does that mean *=0/2?). I don't think you'll like this answer, but finite exact representations of surreals with infinite sign expansions can be written with standard fraction and ordinal notation. "1/3" is a finite exact representation of a surreal with an infinite sign expansion. "$1/\omega +\omega_1/\omega_2$ is a finite exact representation of an uncountably long sign expansion. $\endgroup$
    – Mark S.
    Oct 13 '14 at 2:51
  • $\begingroup$ @MarkS. That's a disappointing answer indeed, but a very clear one too. Good point that $*+*=0 \to *=\frac{0}{2}$ makes little to no sense. And yeah, I guess the fraction representations keeps making sense for all surreals (but not games). Thanks. If you wrap that up in a nice actual answer, I'll accept it. $\endgroup$
    – kram1032
    Oct 13 '14 at 21:32
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Unfortunately, division doesn't really make sense when you step outside the surreals. For example, $*n+*n=0$ for all $n$ (e.g. $*+*=0$), so that $0/2$ would no longer be unique. (As an aside, every game has at least one "half" that you can add to itself to get the original game: see this answer.)

For infinite elements and similar things you can use the fact that a copy of the ordinals sits inside the surreals (albeit with non-standard addition and multiplication).

To represent things in a finite way, you can simply use conventional mathematics notation, like $\dfrac{17}\omega-\dfrac{\omega_{1}^{\mathrm{CK}}}{\omega_2}$, and if it's not obvious that can be dealt with in a computable way because math notation isn't linear, just use $\LaTeX$ or similar, like $\texttt{\dfrac{17}\omega-\dfrac{\omega_{1}^{\mathrm{CK}}}{\omega_2}}$. (I included the Church-Kleene ordinal to play with the "computable" part of your question.)

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