Atomic model (mathematical logic)


In model theory, a subfield of mathematical logic, an atomic model is a model such that the complete type of every tuple is axiomatized by a single formula. Such types are called principal types, and the formulas that axiomatize them are called complete formulas.

Definitions

Let T be a theory. A complete type p is called principal or atomic if it is axiomatized relative to T by a single formula φ ∈ p.
A formula φ is called complete in T if for every formula ψ, the theory T ∪ entails exactly one of ψ and ¬ψ.
It follows that a complete type is principal if and only if it contains a complete formula.
A model M is called atomic if every n-tuple of elements of M satisfies a formula that is complete in Th—the theory of M.

Examples

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.