Editing Surreal number (section)

== Infinity ==

Define {{math|''S''{{sub|''ω''}}}} as the set of all surreal numbers generated by the construction rule from subsets of {{math|''S''{{sub|∗}}}}. (This is the same inductive step as before, since the ordinal number {{mvar|ω}} is the smallest ordinal that is larger than all natural numbers; however, the set union appearing in the inductive step is now an infinite union of finite sets, and so this step can be performed only in a set theory that allows such a union.)  A unique infinitely large positive number occurs in {{math|''S''{{sub|''ω''}}}}:
<math display=block>\omega = \{ S_* \mid{} \} = \{ 1, 2, 3, 4, \ldots \mid{} \}.</math>
{{math|''S''{{sub|''ω''}}}} also contains objects that can be identified as the [[rational number]]s. For example, the {{mvar|ω}}-complete form of the fraction {{sfrac|1|3}} is given by:
<math display=block>\tfrac{1} {3} = \{ y \in S_*: 3 y < 1 \mid y \in S_*: 3 y > 1 \}.</math>
The product of this form of {{sfrac|1|3}} with any form of 3 is a form whose left set contains only numbers less than 1 and whose right set contains only numbers greater than 1; the birthday property implies that this product is a form of 1.

Not only do all the rest of the [[rational number]]s appear in {{math|''S''{{sub|''ω''}}}}; the remaining finite [[real number]]s do too. For example,
<math display=block>\pi = \left\{ 3, \tfrac{25}{8},\tfrac{201}{64}, \ldots \mid 4, \tfrac{7}{2}, \tfrac{13}{4}, \tfrac{51}{16},\ldots \right\}.</math>

The only infinities in {{math|''S''{{sub|''ω''}}}} are {{mvar|ω}} and {{math|−''ω''}}; but there are other non-real numbers in {{math|''S''{{sub|''ω''}}}} among the reals.  Consider the smallest positive number in {{math|''S''{{sub|''ω''}}}}:
<math display=block>\varepsilon = \{ S_- \cup S_0 \mid S_+ \} = \left\{ 0 \mid 1, \tfrac{1}{2}, \tfrac{1}{4}, \tfrac{1}{8}, \ldots \right\} = \{ 0 \mid y \in S_* : y > 0 \}</math>
This number is larger than zero but less than all positive dyadic fractions. It is therefore an [[infinitesimal]] number, often labeled {{mvar|ε}}. The {{mvar|ω}}-complete form of {{mvar|ε}} (respectively {{math|−''ε''}}) is the same as the {{mvar|ω}}-complete form of 0, except that 0 is included in the left (respectively right) set. The only "pure" infinitesimals in {{math|''S''{{sub|''ω''}}}} are {{mvar|ε}} and its additive inverse {{math|−''ε''}}; adding them to any dyadic fraction {{mvar|y}} produces the numbers {{math|1=''y'' ± ''ε''}}, which also lie in {{math|''S''{{sub|''ω''}}}}.

One can determine the relationship between {{mvar|ω}} and {{mvar|ε}} by multiplying particular forms of them to obtain:
: {{math|1=''ω'' · ''ε'' = {{mset| ''ε'' · ''S''{{sub|+}} {{!}} ''ω'' · ''S''{{sub|+}} + ''S''{{sub|∗}} + ''ε'' · ''S''{{sub|∗}} }}}}.
This expression is well-defined only in a set theory which permits transfinite induction up to {{math|''S''{{sub|''ω''{{sup|2}}}}}}.  In such a system, one can demonstrate that all the elements of the left set of {{math|''ωS''{{sub|''ω''}}{{hsp}}·{{px2}}''S''{{sub|''ω''}}''ε''}} are positive infinitesimals and all the elements of the right set are positive infinities, and therefore {{math|''ωS''{{sub|''ω''}}{{hsp}}·{{px2}}''S''{{sub|''ω''}}''ε''}} is the oldest positive finite number, 1.  Consequently, {{math|1={{sfrac|1|''ε''}} = ''ω''}}. Some authors systematically use {{math|''ω''{{sup|−1}}}} in place of the symbol {{mvar|ε}}.

===Contents of ''S''{{sub|''ω''}}===

Given any {{math|1=''x'' = {{mset| ''L'' {{!}} ''R'' }}}} in {{math|''S''{{sub|''ω''}}}}, exactly one of the following is true:
* {{mvar|L}} and {{mvar|R}} are both empty, in which case {{math|1=''x'' = 0}};
* {{mvar|R}} is empty and some integer {{math|''n'' ≥ 0}} is greater than every element of {{mvar|L}}, in which case {{mvar|x}} equals the smallest such integer {{mvar|n}};
* {{mvar|R}} is empty and no integer {{mvar|n}} is greater than every element of {{mvar|L}}, in which case {{mvar|x}} equals {{math|+''ω''}};
* {{mvar|L}} is empty and some integer {{math|''n'' ≤ 0}} is less than every element of {{mvar|R}}, in which case {{mvar|x}} equals the largest such integer {{mvar|n}};
* {{mvar|L}} is empty and no integer {{mvar|n}} is less than every element of {{mvar|R}}, in which case {{mvar|x}} equals {{math|−''ω''}};
* {{mvar|L}} and {{mvar|R}} are both non-empty, and:
** Some dyadic fraction {{mvar|y}} is "strictly between" {{mvar|L}} and {{mvar|R}} (greater than all elements of {{mvar|L}} and less than all elements of {{mvar|R}}), in which case {{mvar|x}} equals the oldest such dyadic fraction {{mvar|y}};
** No dyadic fraction {{mvar|y}} lies strictly between {{mvar|L}} and {{mvar|R}}, but some dyadic fraction <math display=inline> y \in L</math> is greater than or equal to all elements of {{mvar|L}} and less than all elements of {{mvar|R}}, in which case {{mvar|x}} equals {{math|1=''y'' + ''ε''}};
** No dyadic fraction {{mvar|y}} lies strictly between {{mvar|L}} and {{mvar|R}}, but some dyadic fraction <math display=inline> y \in R</math> is greater than all elements of {{mvar|L}} and less than or equal to all elements of {{mvar|R}}, in which case {{mvar|x}} equals {{math|1=''y'' − ''ε''}};
** Every dyadic fraction is either greater than some element of {{mvar|R}} or less than some element of {{mvar|L}}, in which case {{mvar|x}} is some real number that has no representation as a dyadic fraction.

{{math|''S''{{sub|''ω''}}}} is not an algebraic field, because it is not closed under arithmetic operations; consider {{math|''ω'' + 1}}, whose form
<math display=block>\omega + 1 = \{ 1, 2, 3, 4, ... \mid {} \} + \{ 0 \mid{} \} = \{ 1, 2, 3, 4, \ldots, \omega \mid {} \}</math>
does not lie in any number in {{math|''S''{{sub|''ω''}}}}. The maximal subset of {{math|''S''{{sub|''ω''}}}} that is closed under (finite series of) arithmetic operations is the field of real numbers, obtained by leaving out the infinities {{math|±''ω''}}, the infinitesimals {{math|±''ε''}}, and the infinitesimal neighbors {{math|1=''y'' ± ''ε''}} of each nonzero dyadic fraction {{mvar|y}}.

This construction of the real numbers differs from the [[Dedekind cut]]s of [[Real analysis|standard analysis]] in that it starts from dyadic fractions rather than general rationals and naturally identifies each dyadic fraction in {{math|''S''{{sub|''ω''}}}} with its forms in previous generations.  (The {{mvar|ω}}-complete forms of real elements of {{math|''S''{{sub|''ω''}}}} are in one-to-one correspondence with the reals obtained by Dedekind cuts, under the proviso that Dedekind reals corresponding to rational numbers are represented by the form in which the cut point is omitted from both left and right sets.)  The rationals are not an identifiable stage in the surreal construction; they are merely the subset {{mvar|Q}} of {{math|''S''{{sub|''ω''}}}} containing all elements {{mvar|x}} such that {{math|1=''x'' ''b'' = ''a''}} for some {{mvar|a}} and some nonzero {{mvar|b}}, both drawn from {{math|''S''{{sub|∗}}}}.  By demonstrating that {{mvar|Q}} is closed under individual repetitions of the surreal arithmetic operations, one can show that it is a field; and by showing that every element of {{mvar|Q}} is reachable from {{math|''S''{{sub|∗}}}} by a finite series (no longer than two, actually) of arithmetic operations ''including multiplicative inversion'', one can show that {{mvar|Q}} is strictly smaller than the subset of {{math|''S''{{sub|''ω''}}}} identified with the reals.

The set {{math|''S''{{sub|''ω''}}}} has the same [[cardinality]] as the real numbers {{mvar|R}}. This can be demonstrated by exhibiting surjective mappings from {{math|''S''{{sub|''ω''}}}} to the closed unit interval {{mvar|I}} of {{mvar|R}} and vice versa. Mapping {{math|''S''{{sub|''ω''}}}} onto {{mvar|I}} is routine; map numbers less than or equal to {{mvar|ε}} (including {{math|−''ω''}}) to 0, numbers greater than or equal to {{math|1=1 − ''ε''}} (including {{mvar|ω}}) to 1, and numbers between {{mvar|ε}} and {{math|1=1 − ''ε''}} to their equivalent in {{mvar|I}} (mapping the infinitesimal neighbors {{math|''y''±''ε''}} of each dyadic fraction {{mvar|y}}, along with {{mvar|y}} itself, to {{mvar|y}}).  To map {{mvar|I}} onto {{math|''S''{{sub|''ω''}}}}, map the (open) central third ({{sfrac|1|3}}, {{sfrac|2|3}}) of {{mvar|I}} onto {{math|1={{mset| {{!}} }} = 0}}; the central third ({{sfrac|7|9}}, {{sfrac|8|9}}) of the upper third to {{math|1={{mset| 0 {{!}} }} = 1}}; and so forth.  This maps a nonempty open interval of {{mvar|I}} onto each element of {{math|''S''{{sub|∗}}}}, monotonically.  The residue of {{mvar|I}} consists of the [[Cantor set]] {{math|''2''{{sup|''ω''}}}}, each point of which is uniquely identified by a partition of the central-third intervals into left and right sets, corresponding precisely to a form {{math|{{mset| ''L'' {{!}} ''R'' }}}} in {{math|''S''{{sub|''ω''}}}}.  This places the Cantor set in one-to-one correspondence with the set of surreal numbers with birthday {{mvar|ω}}.