Editing Surreal number (section)

===Axiomatic approach===

In another approach to the surreals, given by Alling,<ref name="Alling" />  explicit construction is bypassed altogether. Instead, a set of axioms is given that any particular approach to the surreals must satisfy. Much like the [[Real numbers#Axiomatic approach|axiomatic approach]] to the reals, these axioms guarantee uniqueness [[up to]] isomorphism.

A triple <math display=inline>\langle \mathbb{No}, \mathrm{<}, b \rangle</math> is a surreal number system if and only if the following hold:

* {{math|<}} is a [[total order]] over <math display=inline>\mathbb{No}</math>
* {{mvar|b}} is a function from <math display=inline>\mathbb{No}</math> [[onto]] the class of all ordinals ({{mvar|b}} is called the "birthday function" on <math display=inline>\mathbb{No}</math>).
* Let {{mvar|A}} and {{mvar|B}} be subsets of <math display=inline>\mathbb{No}</math> such that for all {{math|''x'' ∈ ''A''}} and {{math|''y'' ∈ ''B''}}, {{math|''x'' < ''y''}} (using Alling's terminology, {{math|〈 ''A'', ''B'' 〉}} is a "Conway cut" of <math display=inline>\mathbb{No}</math>). Then there exists a unique <math display=inline>z \in \mathbb{No}</math> such that {{math|''b''(''z'')}} is minimal and for all {{math|''x'' ∈ ''A''}} and all {{math|''y'' ∈ ''B''}}, {{math|''x'' < ''z'' < ''y''}}. (This axiom is often referred to as "Conway's Simplicity Theorem".)
* Furthermore, if an ordinal {{mvar|α}} is greater than {{math|''b''(''x'')}} for all {{math|''x'' ∈ ''A'', ''B''}}, then {{math|''b''(''z'') ≤ ''α''}}. (Alling calls a system that satisfies this axiom a "full surreal number system".)

Both Conway's original construction and the sign-expansion construction of surreals satisfy these axioms.

Given these axioms, Alling<ref name="Alling"/> derives Conway's original definition of {{math|≤}} and develops surreal arithmetic.