Editing On Numbers and Games (section)

== Zeroth Part ... On Numbers ==
{{main|Surreal numbers}}
In the Zeroth Part, Chapter 0, Conway introduces a specialized form of [[set (mathematics)|set]] 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, the [[surreal number]]s. The {L|R} notation then resembles the [[Dedekind cut]].

The ordinal <math>\omega</math> is built by [[transfinite induction]]. As with conventional ordinals, <math>\omega+1</math> can be defined. Thanks to the axiomatic definition of subtraction, <math>\omega-1</math> can also be coherently defined: it is strictly less than <math>\omega</math>, and obeys the "obvious" equality <math>(\omega-1)+1=\omega.</math> Yet, it is still larger than any [[natural number]].

The construction enables an entire zoo of peculiar numbers, the surreals, which form a [[field (mathematics)|field]].  Examples include <math>\omega/2</math>, <math>1/\omega</math>, <math>\sqrt{\omega}=\omega^{1/2}</math>, <math>\omega^{1/\omega}</math> and similar.