Editing Surreal number (section)

==Construction==

Surreal numbers are [[inductive definition|constructed inductively]] as [[equivalence class]]es of [[ordered pair|pair]]s of sets of surreal numbers, restricted by the condition that each element of the first set is smaller than each element of the second set. The construction consists of three interdependent parts: the construction rule, the comparison rule and the equivalence rule.

===Forms===
A ''form'' is a pair of sets of surreal numbers, called its ''left set'' and its ''right set''. A form with left set {{mvar|L}} and right set {{mvar|R}} is written {{math|{{mset| ''L'' {{!}} ''R'' }}}}.  When {{mvar|L}} and {{mvar|R}} are given as lists of elements, the braces around them are omitted.

Either or both of the left and right set of a form may be the empty set. The form {{math|{{mset| {{mset| }} {{!}} {{mset| }} }}}} with both left and right set empty is also written {{math|{{mset| {{!}} }}}}.

===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.

===Order===

The recursive definition of surreal numbers is completed by defining comparison:

Given numeric forms {{math|1=''x'' = {{mset| ''X''{{sub|''L''}} {{!}} ''X''{{sub|''R''}} }}}} and {{math|1=''y'' = {{mset| ''Y''{{sub|''L''}} {{!}} ''Y''{{sub|''R''}} }}}}, {{math|''x'' ≤ ''y''}} if and only if both:
*There is no {{math|''x''{{sub|''L''}} ∈ ''X''{{sub|''L''}}}} such that {{math|''y'' ≤ ''x''{{sub|''L''}}}}. That is, every element in the left part of {{mvar|x}} is strictly smaller than {{mvar|y}}.
*There is no {{math|''y''{{sub|''R''}} ∈ ''Y''{{sub|''R''}}}} such that {{math|''y''{{sub|''R''}} ≤ ''x''}}. That is, every element in the right part of {{mvar|y}} is strictly larger than {{mvar|x}}.
Surreal numbers can be compared to each other (or to numeric forms) by choosing a numeric form from its equivalence class to represent each surreal number.

===Induction===

This group of definitions is [[recursion|recursive]], and requires some form of [[mathematical induction]] to define the universe of objects (forms and numbers) that occur in them.  The only surreal numbers reachable via ''finite induction'' are the [[Dyadic rational|dyadic fractions]]; a wider universe is reachable given some form of [[transfinite induction]].

====Induction rule====
* There is a generation {{math|1=''S''{{sub|0}} = {{mset| 0 }}}}, in which 0 consists of the single form {{math|{{mset| {{!}} }}}}.
* Given any [[ordinal number]] {{mvar|n}}, the generation {{math|''S''{{sub|''n''}}}} is the set of all surreal numbers that are generated by the construction rule from subsets of <math display=inline>\bigcup_{i < n} S_i</math>.

The base case is actually a special case of the induction rule, with 0 taken as a label for the "least ordinal".  Since there exists no {{math|''S''{{sub|''i''}}}} with {{math|''i'' < 0}}, the expression <math display=inline>\bigcup_{i < 0} S_i</math> is the empty set; the only subset of the empty set is the empty set, and therefore {{math|''S''{{sub|0}}}} consists of a single surreal form {{math|{{mset| {{!}} }}}} lying in a single equivalence class 0.

For every finite ordinal number {{mvar|n}}, {{math|''S''{{sub|''n''}}}} is [[well-order]]ed by the ordering induced by the comparison rule on the surreal numbers.

The first iteration of the induction rule produces the three numeric forms {{math|{{mset| {{!}} 0 }} < {{mset| {{!}} }} < {{mset| 0 {{!}} }}}} (the form {{math|{{mset| 0 {{!}} 0 }}}} is non-numeric because {{math|0 ≤ 0}}). The equivalence class containing {{nowrap|{{math|{{mset| 0 {{!}} }}}}}} is labeled 1 and the equivalence class containing {{nowrap|{{math|{{mset| {{!}} 0 }}}}}} is labeled −1. These three labels have a special significance in the axioms that define a [[ring (mathematics)|ring]]; they are the additive identity (0), the multiplicative identity (1), and the additive inverse of 1 (−1). The arithmetic operations defined below are consistent with these labels.

For every {{math|''i'' < ''n''}}, since every valid form in {{math|''S''{{sub|''i''}}}} is also a valid form in {{math|''S''{{sub|''n''}}}}, all of the numbers in {{math|''S''{{sub|''i''}}}} also appear in {{math|''S''{{sub|''n''}}}} (as supersets of their representation in {{math|''S''{{sub|''i''}}}}).  (The set union expression appears in our construction rule, rather than the simpler form {{math|''S''{{sub|''n''−1}}}}, so that the definition also makes sense when {{mvar|n}} is a [[limit ordinal]].)  Numbers in {{math|''S''{{sub|''n''}}}} that are a superset of some number in {{math|''S''{{sub|''i''}}}} are said to have been ''inherited'' from generation {{mvar|i}}.  The smallest value of {{mvar|α}} for which a given surreal number appears in {{math|''S''{{sub|''α''}}}} is called its ''birthday''.  For example, the birthday of 0 is 0, and the birthday of −1 is 1.

A second iteration of the construction rule yields the following ordering of equivalence classes:

: {{math|1={{mset| {{!}} −1 }} = {{mset| {{!}} −1, 0 }} = {{mset| {{!}} −1, 1 }} = {{mset| {{!}} −1, 0, 1 }}}}
: {{math|1=< {{mset| {{!}} 0 }} = {{mset| {{!}} 0, 1 }}}}
: {{math|1=< {{mset| −1 {{!}} 0 }} = {{mset| −1 {{!}} 0, 1 }}}}
: {{math|1=< {{mset| {{!}} }} = {{mset| −1 {{!}} }} = {{mset| {{!}} 1 }} = {{mset| −1 {{!}} 1 }}}}
: {{math|1=< {{mset| 0 {{!}} 1 }} = {{mset| −1, 0 {{!}} 1 }}}}
: {{math|1=< {{mset| 0 {{!}} }} = {{mset| −1, 0 {{!}} }}}}
: {{math|1=< {{mset| 1 {{!}} }} = {{mset| 0, 1 {{!}} }} = {{mset| −1, 1 {{!}} }} = {{mset| −1, 0, 1 {{!}} }}}}

Comparison of these equivalence classes is consistent, irrespective of the choice of form. Three observations follow:
# {{math|''S''{{sub|2}}}} contains four new surreal numbers. Two contain extremal forms: {{math|{{mset| {{!}} −1, 0, 1 }}}} contains all numbers from previous generations in its right set, and {{math|{{mset| −1, 0, 1 {{!}} }}}} contains all numbers from previous generations in its left set. The others have a form that partitions all numbers from previous generations into two non-empty sets.
# Every surreal number {{mvar|x}} that existed in the previous "generation" exists also in this generation, and includes at least one new form: a partition of all numbers ''other than'' {{mvar|x}} from previous generations into a left set (all numbers less than {{mvar|x}}) and a right set (all numbers greater than {{mvar|x}}).
# The equivalence class of a number depends on only the maximal element of its left set and the minimal element of the right set.

The informal interpretations of {{math|{{mset| 1 {{!}} }}}} and {{math|{{mset| {{!}} −1 }}}} are "the number just after 1" and "the number just before −1" respectively; their equivalence classes are labeled 2 and −2.  The informal interpretations of {{math|{{mset| 0 {{!}} 1 }}}} and {{math|{{mset| −1 {{!}} 0 }}}} are "the number halfway between 0 and 1" and "the number halfway between −1 and 0" respectively; their equivalence classes are labeled {{sfrac|1|2}} and −{{sfrac|1|2}}.  These labels will also be justified by the rules for surreal addition and multiplication below.

The equivalence classes at each stage {{mvar|n}} of induction may be characterized by their {{mvar|n}}-''complete forms'' (each containing as many elements as possible of previous generations in its left and right sets).  Either this complete form contains ''every'' number from previous generations in its left or right set, in which case this is the first generation in which this number occurs; or it contains all numbers from previous generations but one, in which case it is a new form of this one number. We retain the labels from the previous generation for these "old" numbers, and write the ordering above using the old and new labels:
: {{math|1=−2 < −1 < −{{sfrac|1|2}} < 0 < {{sfrac|1|2}} < 1 < 2}}.

The third observation extends to all surreal numbers with finite left and right sets. (For infinite left or right sets, this is valid in an altered form, since infinite sets might not contain a maximal or minimal element.)  The number {{math|{{mset| 1, 2 {{!}} 5, 8 }}}} is therefore equivalent to {{math|{{mset| 2 {{!}} 5 }}}}; one can establish that these are forms of 3 by using the ''birthday property'', which is a consequence of the rules above.

====Birthday property====
A form {{math|1=''x'' = {{mset| ''L'' {{!}} ''R'' }}}} occurring in generation {{mvar|n}} represents a number inherited from an earlier generation {{math|''i'' < ''n''}} if and only if there is some number in {{math|''S''{{sub|''i''}}}} that is greater than all elements of {{mvar|L}} and less than all elements of the {{mvar|R}}.  (In other words, if {{mvar|L}} and {{mvar|R}} are already separated by a number created at an earlier stage, then {{mvar|x}} does not represent a new number but one already constructed.)  If {{mvar|x}} represents a number from any generation earlier than {{mvar|n}}, there is a least such generation {{mvar|i}}, and exactly one number {{mvar|c}} with this least {{mvar|i}} as its birthday that lies between {{mvar|L}} and {{mvar|R}};  {{mvar|x}} is a form of this {{mvar|c}}. In other words, it lies in the equivalence class in {{math|''S''{{sub|''n''}}}} that is a superset of the representation of {{mvar|c}} in generation {{mvar|i}}.