A question about something in Conway's "On Numbers and Games" 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?
 A: 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.
