Haven't watched the video yet, but then Conway and others formalized a bunch of theory for 2-player combinatorial games like Go, using surreal numbers for game states. (e.g. the number * = {0|0}, or, whoever moves first wins.)
{0|0} = * isn't strictly speaking a surreal number. It is a member of a class that the surreal numbers are a subclass of, called the class of Games.
The definition of a surreal number requires that everything on the left set be less than everything in the right set, but 0 is not less than 0 , so {0|0} does not name a surreal number.
Games do not have this restriction.
The surreal numbers form a field (except that it has a proper class of elements instead of a set of elements) , while the Games do not form a field. The Games do however form a group under addition iirc. Yes, that seems true to me.
The Games represent all perfect information 2 player games with what is iirc called the normal play condition (i.e. you lose iff it is your turn and you have no valid moves), and are such that there is no sequence of moves by the 2 players such that the game never ends.
(Chess for example, is almost a Game , except that it is possible to reach a draw in chess, and the win conditions are somewhat different.
Tic-Tac-Toe with the change that, instead of winning when you have 3 in a row, you win when it is the other player's turn and they have no valid moves, and you cannot move if the other player has 3 in a row, is a Game in this sense.)
Note: I think that Games is often capitalized because of it being a proper class, but I am not sure.
