nLab DLO

Redirected from "dense linear order".
Contents

Context

Model theory

Analysis

Contents

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 DLO\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

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

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

  • Since DLO\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.

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

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

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

  • Let (A,<)(A,\lt) be a model of DLO\mathsf{DLO}. Then AA 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.

References

Last revised on July 5, 2022 at 16:01:38. See the history of this page for a list of all contributions to it.