Editing Surreal number (section)

===Consistency===
It can be shown that the definitions of negation, addition and multiplication are consistent, in the sense that:
* Addition and negation are defined recursively in terms of "simpler" addition and negation steps, so that operations on numbers with birthday {{mvar|n}} will eventually be expressed entirely in terms of operations on numbers with birthdays less than {{mvar|n}};
* Multiplication is defined recursively in terms of additions, negations, and "simpler" multiplication steps, so that the product of numbers with birthday {{mvar|n}} will eventually be expressed entirely in terms of sums and differences of products of numbers with birthdays less than {{mvar|n}};
* As long as the operands are well-defined surreal number forms (each element of the left set is less than each element of the right set), the results are again well-defined surreal number forms;
* The operations can be extended to ''numbers'' (equivalence classes of forms): the result of negating {{mvar|x}} or adding or multiplying {{mvar|x}} and {{mvar|y}} will represent the same number regardless of the choice of form of {{mvar|x}} and {{mvar|y}}; and
* These operations obey the associativity, commutativity, additive inverse, and distributivity axioms in the definition of a [[field (mathematics)|field]], with additive identity {{math|1=0 = {{mset| {{!}} }}}} and multiplicative identity {{math|1=1 = {{mset| 0 {{!}} }}}}.

With these rules one can now verify that the numbers found in the first few generations were properly labeled.  The construction rule is repeated to obtain more generations of surreals:

: {{math|1=''S''{{sub|0}} = {{mset| 0 }}}}
: {{math|1=''S''{{sub|1}} = {{mset| −1 < 0 < 1 }}}}
: {{math|1=''S''{{sub|2}} = {{mset| −2 < −1 < −{{sfrac|1|2}} < 0 < {{sfrac|1|2}} < 1 < 2 }}}}
: {{math|1=''S''{{sub|3}} = {{mset| −3 < −2 < −{{sfrac|3|2}} < −1 < −{{sfrac|3|4}} < −{{sfrac|1|2}} < −{{sfrac|1|4}} < 0 < {{sfrac|1|4}} < {{sfrac|1|2}} < {{sfrac|3|4}} < 1 < {{sfrac|3|2}} < 2 < 3 }}}}
: {{math|1=''S''{{sub|4}} = {{mset| −4 < −3 < ... < −{{sfrac|1|8}} < 0 < {{sfrac|1|8}} < {{sfrac|1|4}} < {{sfrac|3|8}} < {{sfrac|1|2}} < {{sfrac|5|8}} < {{sfrac|3|4}} < {{sfrac|7|8}} < 1 < {{sfrac|5|4}} < {{sfrac|3|2}} < {{sfrac|7|4}} < 2 < {{sfrac|5|2}} < 3 < 4 }}}}