 Dense order

A total order is dense when between every two distinct ordered points lies a third. Endpoints are allowed; “without endpoints” is an additional condition. A dense order with two points $a<b$ cannot be a well-order, since $\{x:a<x\}$ is nonempty and has no least member. Its first-order axioms are the total-order axioms plus $\forall x,y(x<y\Rightarrow \exists z(x<z<y))$. Countability and uncountability are not first-order axiomatizable restrictions, by the upward and downward witness constructions of the Löwenheim-Skolem theorem.