# Mistake in ONAG?

In the second edition of the book "On Numbers and Games" by Conway there is a theorem 88 (p. 194) on comparison of sums of ${\uparrow} x$ games. It contains a weird statement:

... (If $X$ is a sum of ${\uparrow} x$'s, then) The game $X + *$ is positive if and only if $X + {\uparrow} \mathbf{On}$ is positive, where $\mathbf{On}$ is a number larger than any of $x$'s among the summands of $X$ (definition slightly abridged)

This statement doesn't really make any sense, since it implies $0 + *$ is positive (which it isn't), and can even apply to some negative $X$ (e.g. $X = {\downarrow}$). I feel like this is a mistake, and $X + {\uparrow} \mathbf{On}$ should be replaced with $X + {\downarrow} \mathbf{On}$. After this replacement it sounds a lot like truth. Can anyone please confirm, or input on this?

• Your proposed correction is consistent with the reformulation on the next page, which says that * is confused with all values between $\downarrow$On and $\uparrow$On, greater than smaller values, and less than larger values. – Hugh Thomas Sep 11 '17 at 17:14
• Yes; with this reformulation being stated explicitly it seems that the speculated meaning is what was intended, and the theorem simply contains a typo. It is hard to be sure without a proof though. – Mikhail Tikhomirov Sep 11 '17 at 17:24
• If there were proofs, it wouldn't be ONAG... – Hugh Thomas Sep 11 '17 at 18:23