 Solution

ID: past-exam-of-the-logic-and-set-theory-course-of-the-university-of-oxford/2010/a/3/d/i/solution

\forall x \neg (x < x), \forall x \forall y \forall z ((x < y \land y < z) \to x < z),

\forall x \forall y (x < y \lor x = y \lor y < x), \forall x \forall y (x < y \to \exists z (x < z \land z < y)),

\forall x \exists y (y < x), \forall x \exists y (x < y) .

The first three give strict linear order; the next gives density; the final two exclude endpoints. Domains are nonempty under the adopted first-order semantics, and these axioms force them to be infinite.

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