## Recent additions[edit]

I am not sure that all my recent additions to the navbox are helpful. I await feedback before adding the navbox to the article pages of the articles that I linked from the navbox. —Mark Dominus (talk) 21:05, 3 February 2011 (UTC)

- They are all fine. Perhaps a separate group "controversies" could include berkeley's criticism and the page on criticism of nsa. Tkuvho (talk) 04:10, 4 February 2011 (UTC)

## Games ✶ and ↑[edit]

At some point the navbox should mention games like ✶ and ↑, which are infinitesimal in a rather different sense than the other infinitesimals mentioned in the navbox. But I don't think our coverage of these is clear enough at present for it to be worth while to add them now. —Mark Dominus (talk)

- I don't really understand them. Perhaps if there were a decent explanation somewhere, I would be more enthusiastic. Is this related to Hackenbush strings? Not that I understand the latter. Tkuvho (talk) 10:36, 8 February 2011 (UTC)

- Are you familiar with the definition of the surreal numbers? Briefly, a surreal number
*n*is an ordered pair (*N*,_{L}*N*) of sets of surreal numbers, such that there is no element of_{R}*N*is ≤ any element of_{R}*N*, for a suitably-defined ≤ relation. Then ({}, {}) plays the role of the number 0, (, {}) and ({}, ) become +1 and -1, (, ) is +½, and so forth._{L}

- Are you familiar with the definition of the surreal numbers? Briefly, a surreal number

- If one drops the ≤ condition on the two sets, one gets the more general class of "games": a game is simply an ordered pair of sets of games. Clearly all numbers are games. But there are non-numeric games, such as ✶, which is defined as (, ). This fails the ≤ condition, because 0≤0, so it is a non-number game. If one uses the definition of ≤ to compare ✶ with various numbers, one finds that -
*p*< ✶ <*p*for all positive numbers*p*, but that neither ✶ ≤ 0 nor 0 ≤ ✶ holds. < is a total ordering on numbers, but not on games.

- If one drops the ≤ condition on the two sets, one gets the more general class of "games": a game is simply an ordered pair of sets of games. Clearly all numbers are games. But there are non-numeric games, such as ✶, which is defined as (, ). This fails the ≤ condition, because 0≤0, so it is a non-number game. If one uses the definition of ≤ to compare ✶ with various numbers, one finds that -

- Similarly, ↑ is defined to be the non-number game (, {✶}), and one can show that although ↑ is positive (that is, 0 ≤ ↑ but not ↑ ≤ 0), ↑ <
*p*for all positive numbers*p*. This has real implications for combinatorial game theory: games that are positive numbers behave as advantages of various sizes for the "Left" player, and negative numbers are the corresponding advantages for the "Right" player. One can determine the winner of a position by adding up the numbers; if the sum is positive, Left can force a win, and if negative, Right can force a win. The game ↑ is an advantage for Left, but only an infinitesimal one, and not enough to outweigh any numeric advantage for Right, however small.

- Similarly, ↑ is defined to be the non-number game (, {✶}), and one can show that although ↑ is positive (that is, 0 ≤ ↑ but not ↑ ≤ 0), ↑ <

- As I said, Wikipedia's present coverage of these topics is both thin and disorganized, and I would not want to add them to the navbox until the coverage improves. —Mark Dominus (talk) 16:37, 8 February 2011 (UTC)

- Thanks. Incidentally, it may be useful to develop a law of continuity page starting with Leibniz. Tkuvho (talk) 08:02, 9 February 2011 (UTC)

## Dehn[edit]

The page Dehn planes was linked here but was de-linked by a user on the grounds that they "do not use infinitesimals in this sense". The user did not respond to a request for clarification. I suggest we re-instate the link to Dehn planes. Tkuvho (talk) 19:38, 20 March 2011 (UTC)