It uses sets to define a universe of numbers (whole numbers, fractional numbers and some very strange surreal numbers of no use to the general public) and as a byproduct, uses the same method to define some "games".
At the start the only set you have is the empty set (ie the set with no members). The number 0 is defined as the result of comparing the empty set with the empty set (written as {} ). Now you can compare the set comprising 0 with the empty set, {0}, which is results in the number 1; you can compare the empty set with 0, {0}, which is results in 1; and you can compare 0 with 0, {00}, which is the first "game" and is called * (star).
Star is not a "game" as such but is the value for a common position in many games. In a game played between Left and Right, if the position has the value 1 (a number) then it is a win for Right. If 1 a win for Left, if 0 a win for the second person to play; and if * a win for the first player.
The book is in two, (0,1}, parts. The zeroth part is about numbers, the first part about games  both the values of games and also some real games that can be played such as Nim, Hackenbush[?], Col[?] and Snort[?] amongst the many described.
For a more detailed treatment of the mathematics involved, see surreal numbers.
See also: Winning Ways for your Mathematical Plays.
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