===== Rational numbers =====
==== Framework ====
$\dots,\,-\frac{4}{3},\,0,\,\frac{1}{17},\,1,\,7.528,\,9001,\dots$

The rational numbers ${\mathbb{Q}}$ can be defined as the field of characteristic 0 which has no proper sub-field. In less primitive notions, it's the field of fractions for the integral domain of [[natural numbers]]. The second order theory of rationals (see the note below) describes a countable collection.

The rationals can also be set up straight forwardly from tuples of natural numbers. 

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=== Discussion ===
== Theorems ==
For all $m$

^ $\dfrac{1}{1-x}=\dfrac{1}{1-x\cdot x^{m}}\sum_{k=0}^m x^k$ ^
^ $\dfrac{1}{y}=\dfrac{1}{1-(1-y)\cdot(1-y)^{m}}\sum_{k=0}^m (1-y)^k$ ^

== Logic ==
In first order logic, being of characteristic zero ("$\forall n.\,(1+1+\dots+1)_{n\ \text{times}}\neq 0$") requires an axiom schema. But even the induction axiom of the Peano axioms requires a schema.

Also, due to the Löwenheim–Skolem theorem, //all// theories of infinite structures (e.g. ${\mathbb N}, {\mathbb Q}, {\mathbb R}$) have bad properties in first order logic. For $\mathbb Q$, there is a first-order theory of fields and one can also characterize characteristic 0, however the notion of "proper subfield" is elusive. One needs second-order logical to capture it categorically (=all possible models are isomorphic), see the references.

=== References ===
Wikipedia: 
[[http://en.wikipedia.org/wiki/Field_of_fractions|Field of fractions]],
[[http://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem|Löwenheim–Skolem theorem]]

Math StackExchange: 
[[http://math.stackexchange.com/questions/262329/axiomatic-characterization-of-the-rational-numbers|Axiomatic characterization of the rational numbers]], 
[[http://math.stackexchange.com/questions/1082604/what-is-the-first-order-axiom-characterizing-a-field-having-characteristic-zero|What is the first order axiom characterizing a field having characteristic zero?]]

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=== Related ===
[[Logic]], [[Real numbers]]