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.

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: Field of fractions, Löwenheim–Skolem theorem

Math StackExchange: Axiomatic characterization of the rational numbers, What is the first order axiom characterizing a field having characteristic zero?

Logic, Real numbers