4
$\begingroup$

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?

$\endgroup$
  • 1
    $\begingroup$ 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. $\endgroup$ – Hugh Thomas Sep 11 '17 at 17:14
  • 1
    $\begingroup$ 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. $\endgroup$ – Mikhail Tikhomirov Sep 11 '17 at 17:24
  • 2
    $\begingroup$ If there were proofs, it wouldn't be ONAG... $\endgroup$ – Hugh Thomas Sep 11 '17 at 18:23

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.