Atomic model (mathematical logic)

For models of the atom, see atomic theory.

In model theory, an atomic model is a model which is in some sense small.

Definitions

A formula φ(x₁,...,x_n) in a complete theory T is called complete if for every other formula ψ(x₁,...,x_n), the formula φ implies exactly one of ψ and ¬ψ in T.

A model M of the theory is called atomic if every n-tuple of elements of M satisfies a complete formula.

Properties

The back and forth method can be used to show that any two countable atomic models of a theory that are elementarily equivalent are isomorphic.

References

Chang, Chen Chung; Keisler, H. Jerome (1990) [1973], Model Theory, Studies in Logic and the Foundations of Mathematics (3rd ed.), Elsevier, ISBN 978-0-444-88054-3