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In the book mentioned in the title, which deals with (among other things), Conway's "surreal numbers", there is a small section (pp. 37-38) where the "gaps" in the surreal number line are discussed. The gaps in the number line are formed by Dedekind cuts between whole proper classes of surreal numbers, in particular where we partition the surreal number line into two classes like the cut construction of the real numbers from the rational numbers. It is mentioned also that these gaps (presumably, since proper classes cannot be members in NBG set theory and/or the collection would be "too big") cannot be collected into a whole.

He mentions the existence of a gap "between 0 and all positive numbers", denoting it by $\frac{1}{\mathbf{On}}$, where "$\mathbf{On}$" is the "gap at the end of the number line", given by the (improper?) Dedekind cut where the left class is all of $\mathbf{No}$ and the right class is empty. This is much like the points at infinity on the extended real number line, though Conway uses the symbol $\infty$ for a different gap despite this analogy. But that is where I'm hung: how can there be a gap between 0 and all positive numbers? A Dedekind cut, as far as I can tell, represents a greatest lower or least upper bound of a set, or, in this case, of a (proper) class. Yet the class of "nonpositive surreal numbers" has supremum 0 and the class of "positive surreal numbers" has infimum 0. So the cut just corresponds to 0. So in this case there appears to be no true "gap", i.e. something missing, much less something missing between all positive and nonpositive surreals. Why does he say there is one?

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  • $\begingroup$ Not entirely sure of the details but from what I remember from ONAG, you construct numbers by the $\{ * | * \}$ formalism, and you can basically put whatever you want on the left and the right, e.g. here $\{ 0 | \text{ all } x > 0 \}$. You have created a gap by creating this number -- unlike Dedekind, there's no sense of getting to 0 by getting infinitely close, since the two are different sets constructed differently. $\endgroup$ – Scaramouche Jul 31 '13 at 2:10
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    $\begingroup$ @Scaramouche This is asking about something else. On pp. 37–38 Conway, having constructed the totally ordered set $\mathbf{No}$, then constructs cuts of this set analogous to the way the reals are constructed from the totally ordered rationals by Dedekind cuts. $\endgroup$ – MJD Jul 31 '13 at 3:06
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Even though the infimum and supremum among numbers are equal, there are still games which are inside the gap. See Chapter 16 of ONAG (particularly, the section "Games in the Gaps").

For example, the games $+_{\alpha} = \{0|\{0|-\alpha\}\}$, where $\alpha$ is any positive surreal number, lie in the gap $1/On$. We can check this. The game $+_{\alpha}$ is positive because Left wins no matter who moves first. On the other hand, if $\beta$ is any positive surreal number, consider the game $+_{\alpha} - \beta$. Since $\beta$ is a number and $+_{\alpha}$ is not, neither player wants to move in $\beta$ (Number Avoidance Theorem), so we can just check moves in $+_{\alpha}$. The component $+_{\alpha}$ quickly reduces to 0, leaving $-\beta$, which is a win for Right. Therefore, $+_{\alpha} < \beta$.

(The Number Avoidance Theorem is not explicitly stated as a theorem in ONAG, but is essentially the comment labelled "Summary" in Chapter 9, in the section "Stopping Positions".)

The same argument shows that any positive all-small game lies in the gap $1/On$. An all-small game is a game having the property that if one player has legal moves, then the other one does too. Equivalently, the only possible number that can be reached while playing the game is 0. (See Chapter 9 of ONAG.) Then the same argument as above applies.

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  • $\begingroup$ So if I get this right, Conway is apparently interpreting a Dedekind cut in a slightly different way than how it is interpreted when constructing the reals. In particular, interpreting it as always indicating a gap between the left and right class, even if the left or right class has a greatest or least (respectively) element. Is this right? $\endgroup$ – The_Sympathizer Jul 31 '13 at 5:25
  • $\begingroup$ @mike4ty4 I believe that is correct. But some gaps even do not have any games in it, as mentioned at the beginning of the "Games in the Gaps" section. I don't totally understand this but I think he's saying that in order for a gap to contain games, the left and right sections must be sets (as opposed to proper classes). Conway gives some examples of this on pp. 37-38. $\endgroup$ – Ted Jul 31 '13 at 6:22
  • $\begingroup$ Correct me if I'm wrong, but if the left and right sections are sets of surreals, and nothing in the right set is less than or equal to anything in the left set, then wouldn't you have a Surreal. I thought a gap was when you have a proper class... $\endgroup$ – Mark S. Aug 2 '13 at 3:27
  • $\begingroup$ @MarkS. You're right; I did not state that properly. The statement in ONAG is that in order for there to be games in a gap, the gap must be the "upper or lower bound of sets." I don't completely understand this, but if you look at Thm 56, this expresses the left and right sections of a game $G$ as the upper (resp., lower) of bound of sections of $G^L$ (resp., $G^R$), and $G^L$ and $G^R$ are sets of numbers. Whereas, if you take the example on p.38 of $\sum_{\alpha} {\omega}^{-\alpha}$ where $\alpha$ ranges over all ordinals, then ... $\endgroup$ – Ted Aug 2 '13 at 5:34
  • $\begingroup$ it would only be the upper bound of all the partial sums $\sum_{\beta<\alpha} \omega^{-\beta}$ (over all ordinals $\alpha$), and there are too many such sums to be a set. $\endgroup$ – Ted Aug 2 '13 at 5:36

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