# nLab omega-categorical structure

model theory

## Dimension, ranks, forking

• forking and dividing?

• Morley rank?

• Shelah 2-rank?

• Lascar U-rank?

• Vapnik–Chervonenkis dimension?

# Contents

## Idea

$\omega$-categorical structures are “highly symmetric” first-order structures.

## Definition

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 $A$.

## Examples

• 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.

## Remarks

• 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.

## References

Created on March 8, 2017 at 02:38:27. See the history of this page for a list of all contributions to it.