In model theory, 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.
Contents
Definitions
A complete type p(x1, ..., xn) is called principal (or atomic) if it is axiomatized by a single formula φ(x1, ..., xn) ∈ p(x1, ..., xn).
A formula φ in a complete theory T is called complete if for every other formula ψ(x1, ..., xn), the formula φ implies exactly one of ψ and ¬ψ in T. It follows that a complete type is principal if and only if it contains a complete formula.
A model M of the theory is called atomic if every n-tuple of elements of M satisfies a complete formula.
Examples
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.