nLab DLO

Idea

Countable dense linear orders without endpoints are unique up to isomorphism, and are canonically modeled by the rational numbers $(\mathbb{Q}, \lt )$.

Definition

The theory $\mathsf{DLO}$ of the dense linear order without endpoints is the first-order theory of $(\mathbb{Q}, \lt)$. It is axiomatized by the usual axioms of a linear order, plus the sentences which state that the order is dense and that there is neither an upper nor lower bound on the order.

Remarks

• $\mathsf{DLO}$ is a prototypical unstable structure.

• Cantor’s theorem (the uniqueness up to isomorphism of a model of $\mathsf{DLO}$ assuming the model is countable) says precisely that $\mathsf{DLO}$ is an omega-categorical theory.

• Since $\mathsf{DLO}$ is unstable, however, its uncountable models fall into many isomorphism classes.

• Dedekind cuts arise as types over bounded infinite parameter sets in a single variable.

• $\mathsf{DLO}$ is a Fraïssé limit; its finitely-generated substructures are precisely the finite linear orders.

• $\mathsf{DLO}$ admits quantifier-elimination.

• If we view $(\mathbb{Q},<)$ as a category, the subobject classifier of the topos $\mathbf{Sets}^\mathbb{Q}$ can be identified in a canonical way with the Dedekind cuts on $\mathbb{Q}$.

• Let $(A,\lt)$ be a model of $\mathsf{DLO}$. Then $A$ has a frame of open subsets with respect to its linear order. When regarded as a locale, the frame of open subsets is isomorphic to the locale of real numbers.

Related concepts

• dense

• omega-categorical structure

• homogeneous structure?

• o-minimal structure

• locator

References

• Ivan Tomasic, Model theory