indiscernible sequence?
Morley sequence?
Ramsey theorem?
Erdos-Rado theorem?
Ehrenfeucht-Fraïssé games (back-and-forth games)
Hrushovski construction?
generic predicate?
analysis (differential/integral calculus, functional analysis, topology)
metric space, normed vector space
open ball, open subset, neighbourhood
convergence, limit of a sequence
compactness, sequential compactness
continuous metric space valued function on compact metric space is uniformly continuous
…
…
Countable dense linear orders without endpoints are unique up to isomorphism, and are canonically modeled by the rational numbers $(\mathbb{Q}, \lt )$.
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.
$\mathsf{DLO}$ 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.
homogeneous structure?
Last revised on July 5, 2022 at 12:01:38. See the history of this page for a list of all contributions to it.