Editing Surreal number (section)

===Numeric forms and their equivalence classes===

'''Construction rule'''
:A form {{math|{{mset| ''L'' {{!}} ''R'' }}}} is ''numeric'' if the intersection of {{mvar|L}} and {{mvar|R}} is the empty set and each element of {{mvar|R}} is greater than every element of {{mvar|L}}, according to the [[order theory|order relation]] ≤ given by the comparison rule below.

The numeric forms are placed in equivalence classes; each such equivalence class is a ''surreal number''. The elements of the left and right sets of a form are drawn from the universe of the surreal numbers (not of ''forms'', but of their ''equivalence classes'').

'''Equivalence rule'''
: Two numeric forms {{mvar|x}} and {{mvar|y}} are forms of the same number (lie in the same equivalence class) if and only if both {{math|''x'' ≤ ''y''}} and {{math|''y'' ≤ ''x''}}.

An [[Order theory|ordering relationship]] must be [[antisymmetric relation|antisymmetric]], i.e., it must have the property that {{math|1=''x'' = ''y''}} (i. e., {{math|''x'' ≤ ''y''}} and {{math|''y'' ≤ ''x''}} are both true) only when {{mvar|x}} and {{mvar|y}} are the same object.  This is not the case for surreal number ''forms'', but is true by construction for surreal ''numbers'' (equivalence classes).

The equivalence class containing {{math|{{mset| {{!}} }}}} is labeled 0; in other words, {{math|{{mset| {{!}} }}}} is a form of the surreal number 0.