indiscernible sequence?
Morley sequence?
Ramsey theorem?
Erdos-Rado theorem?
Ehrenfeucht-Fraïssé games (back-and-forth games)
$\omega$-categorical structures are “highly symmetric” first-order structures.
A countable structure $A$ is $\omega$-categorical if any other countable structure $B \models \mathbf{Th}(A)$ with the same first-order theory as $A$ is isomorphic to $A$.
The countable dense linear orders without endpoints are $\omega$-categorical.
The countable random graph is $\omega$-categorical.
Many Fraisse limits are $\omega$-categorical.
The canonical (orbit) structure induced by an oligomorphic (finitely many orbits in each power) permutation group on a countable set is $\omega$-categorical.
Conversely, the action $\Aut(A) \curvearrowright A$ is oligomorphic.
The automorphism group of an $\omega$-categorical structure $A$ equipped with the topology of pointwise convergence comprises a complete set of invariants for $\mathbf{Th}(A)$ up to bi-interpretability: this is the Coquand-Ahlbrandt-Ziegler theorem.
Created on March 8, 2017 at 02:39:11. See the history of this page for a list of all contributions to it.