On Numbers and Games

First edition
Author	John Horton Conway
Language	English
Genre	Mathematics
Publisher	Academic Press, Inc.
Publication place	United States
Media type	Print
Pages	238 pp.
ISBN	0-12-186350-6

On Numbers and Games is amathematics book byJohn Horton Conway first published in 1976.^[1] The book is written by a pre-eminent mathematician, and is directed at other mathematicians. The material is, however, developed in a playful and unpretentious manner and many chapters are accessible to non-mathematicians.Martin Gardner discussed the book at length, particularly Conway's construction ofsurreal numbers, in hisMathematical Games column inScientific American in September 1976.^[2]

The book is roughly divided into two sections: the first half (orZeroth Part), onnumbers, the second half (orFirst Part), ongames. In theZeroth Part, Conway providesaxioms for arithmetic: addition, subtraction, multiplication, division and inequality. This allows anaxiomatic construction of numbers andordinal arithmetic, namely, theintegers,reals, thecountable infinity, and entire towers of infiniteordinals. The object to which these axioms apply takes the form {L|R}, which can be interpreted as a specialized kind ofset; a kind of two-sided set. By insisting that L<R, this two-sided set resembles theDedekind cut. The resulting construction yields afield, now called thesurreal numbers. The ordinals are embedded in this field. The construction is rooted inaxiomatic set theory, and is closely related to theZermelo–Fraenkel axioms. In the original book, Conway simply refers to this field as "the numbers". The term "surreal numbers" is adopted later, at the suggestion ofDonald Knuth.

In theFirst Part, Conway notes that, by dropping the constraint that L<R, the axioms still apply and the construction goes through, but the resulting objects can no longer be interpreted as numbers. They can be interpreted as theclass of all two-player games. The axioms forgreater than andless than are seen to be a natural ordering on games, corresponding to which of the two players may win. The remainder of the book is devoted to exploring a number of different (non-traditional, mathematically inspired) two-player games, such asnim,hackenbush, and themap-coloring gamescol andsnort. The development includes their scoring, a review of theSprague–Grundy theorem, and the inter-relationships to numbers, including their relationship toinfinitesimals.

The book was first published byAcademic Press in 1976,ISBN 0-12-186350-6, and a second edition was released byA K Peters in 2001 (ISBN 1-56881-127-6).

In the Zeroth Part, Chapter 0, Conway introduces a specialized form ofset notation, having the form {L|R}, where L and R are again of this form, built recursively, terminating in {|}, which is to be read as an analog of the empty set. Given this object, axiomatic definitions for addition, subtraction, multiplication, division and inequality may be given. As long as one insists that L<R (with this holding vacuously true when L or R are the empty set), then the resulting class of objects can be interpreted as numbers, thesurreal numbers. The {L|R} notation then resembles theDedekind cut.

The ordinal${\displaystyle \omega }$ is built bytransfinite induction. As with conventional ordinals,${\displaystyle \omega +1}$ can be defined. Thanks to the axiomatic definition of subtraction,${\displaystyle \omega -1}$ can also be coherently defined: it is strictly less than${\displaystyle \omega }$, and obeys the "obvious" equality${\displaystyle (\omega -1)+1=\omega .}$ Yet, it is still larger than anynatural number.

The construction enables an entire zoo of peculiar numbers, the surreals, which form afield. Examples include${\displaystyle \omega /2}$,${\displaystyle 1/\omega }$,${\displaystyle \sqrt{\omega }={\omega }^{1/2}}$,${\displaystyle {\omega }^{1/\omega }}$ and similar.

In the First Part, Conway abandons the constraint that L<R, and then interprets the form {L|R} as a two-player game: a position in a contest between two players,Left andRight. Each player has aset of games calledoptions to choose from in turn. Games are written {L|R} where L is the set ofLeft's options and R is the set ofRight's options.^[3] At the start there are no games at all, so theempty set (i.e., the set with no members) is the only set of options we can provide to the players. This defines the game {|}, which is called0. We consider a player who must play a turn but has no options to have lost the game. Given this game 0 there are now two possible sets of options, the empty set and the set whose only element is zero. The game {0|} is called 1, and the game {|0} is called -1. The game {0|0} is called* (star), and is the first game we find that is not a number.

All numbers arepositive, negative, or zero, and we say that a game is positive ifLeft has a winning strategy, negative ifRight has a winning strategy, or zero if the second player has a winning strategy. Games that are not numbers have a fourth possibility: they may befuzzy, meaning that the first player has a winning strategy. * is a fuzzy game.^[4]

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