nLab
DLO

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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.

omega-categorical structure

homogeneous structure?

References

Last revised on May 27, 2021 at 13:00:48. See the history of this page for a list of all contributions to it.