Editing Surreal number (section)

== Transfinite induction ==

Continuing to perform [[transfinite induction]] beyond {{math|''S''{{sub|''ω''}}}} produces more ordinal numbers {{mvar|α}}, each represented as the largest surreal number having birthday {{mvar|α}}.  (This is essentially a definition of the ordinal numbers resulting from transfinite induction.)  The first such ordinal is {{math|1=''ω'' + 1 = {{mset| ''ω'' {{!}} }}}}.  There is another positive infinite number in generation {{math|''ω'' + 1}}:
: {{math|1=''ω'' − 1 = {{mset| 0, 1, 2, 3, 4, ... {{!}} ''ω'' }}}}.
The surreal number {{math|1=''ω'' − 1}} is not an ordinal; the ordinal {{math|''ω''}} is not the successor of any ordinal. This is a surreal number with birthday {{math|''ω'' + 1}}, which is labeled {{math|1=''ω'' − 1}} on the basis that it coincides with the sum of {{math|''ω'' {{=}} {{mset| 0, 1, 2, 3, 4, ... {{!}} }}}} and {{math|−1 {{=}} {{mset|  {{!}} 0 }}}}. Similarly, there are two new infinitesimal numbers in generation {{math|1=''ω'' + 1}}:
: {{math|1=2''ε'' = ''ε'' + ''ε'' = {{mset| ''ε'' {{!}} 1 + ''ε'', {{sfrac|1|2}} + ''ε'', {{sfrac|1|4}} + ''ε'', {{sfrac|1|8}} + ''ε'', ... }}}} and
: {{math|1={{sfrac|''ε''|2}} = ''ε'' · {{sfrac|1|2}} = {{mset| 0 {{!}} ''ε'' }}}}.

At a later stage of transfinite induction, there is a number larger than {{math|1=''ω'' + ''k''}} for all natural numbers {{math|''k''}}:
: {{math|1=2''ω'' = ''ω'' + ''ω'' = {{mset| ''ω'' + 1, ''ω'' + 2, ''ω'' + 3, ''ω'' + 4, ... {{!}} }}}}
This number may be labeled {{math|1=''ω'' + ''ω''}} both because its birthday is {{math|1=''ω'' + ''ω''}} (the first ordinal number not reachable from {{math|''ω''}} by the successor operation) and because it coincides with the surreal sum of {{math|''ω''}} and {{math|''ω''}}; it may also be labeled {{math|2''ω''}} because it coincides with the product of {{math|''ω'' {{=}} {{mset| 1, 2, 3, 4, ... {{!}}  }}}} and {{math|2 {{=}} {{mset| 1 {{!}}   }}}}. It is the second limit ordinal; reaching it from {{math|''ω''}} via the construction step requires a transfinite induction on
<math display=block>\bigcup_{k < \omega} S_{\omega + k}</math>
This involves an infinite union of infinite sets, which is a "stronger" set theoretic operation than the previous transfinite induction required.

Note that the ''conventional'' addition and multiplication of ordinals does not always coincide with these operations on their surreal representations.  The sum of ordinals {{math|1=1 + ''ω''}} equals {{math|''ω''}}, but the surreal sum is commutative and produces {{math|1=1 + ''ω'' = ''ω'' + 1 > ''ω''}}. The addition and multiplication of the surreal numbers associated with ordinals coincides with the [[ordinal arithmetic#Natural operations|natural sum and natural product]] of ordinals.

Just as {{math|2''ω''}} is bigger than {{math|1=''ω'' + ''n''}} for any natural number {{math|''n''}}, there is a surreal number {{math|{{sfrac|''ω''|2}}}} that is infinite but smaller than {{math|1=''ω'' − ''n''}} for any natural number {{math|''n''}}. That is, {{math|{{sfrac|''ω''|2}}}} is defined by
: {{math|1={{sfrac|''ω''|2}} = {{mset| ''S''{{sub|∗}} {{!}} ''ω'' − ''S''{{sub|∗}} }}}}
where on the right hand side the notation {{math|''x'' − ''Y''}} is used to mean {{math|{{mset| ''x'' − ''y'' : ''y'' ∈ ''Y'' }}}}.  It can be identified as the product of {{math|''ω''}} and the form {{math|{{mset| 0 {{!}} 1 }}}} of {{math|{{sfrac|1|2}}}}. The birthday of {{math|{{sfrac|''ω''|2}}}} is the limit ordinal {{math|''ω''2}}.