We can uniquely define the set of real numbers as a complete ordered field. But can we do something similar with the set of rational numbers ? I think we need to change the completness axiom with some other one. I know that we can define integers in ordered field so maybe this axiom should state that every number is quotient of two integers ? Or maybe we can choose some simpler one. So what axiom/axioms should we choose instead of completness axiom to define rational numbers ? Thanks in advance.
We can define the field of rational numbers as the smallest ordered field.
More precisely, let $F$ be an ordered field without subfields (except for $F$ itself). Then it is not hard to prove that
- $\mathbb Q$ is one such field;
- every such field is isomorphic to $\mathbb Q$.