Atomic model (mathematical logic)

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.

Definitions

A complete type p(x₁, ..., x_n) is called principal (or atomic) if it is axiomatized by a single formula φ(x₁, ..., x_n) ∈ p(x₁, ..., x_n).

A formula φ 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.^[1] 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

• The ordered field of real algebraic numbers is the unique atomic model of the theory of real closed fields.
• Any finite model is atomic
• A dense linear ordering without endpoints is atomic.
• Any prime model of a countable theory is atomic by the omitting types theorem.
• Any countable atomic model is prime, but there are plenty of atomic models that are not prime, such as an uncountable dense linear order without endpoints.
• The theory of a countable number of independent unary relations is complete but has no completable formulas and no atomic models.

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.

Notes

1. Some authors refer to complete formulas as "atomic formulas", but this is inconsistent with the purely syntactical notion of an atom or atomic formula as a formula that does not contain a proper subformula.

References

• Chang, Chen Chung; Keisler, H. Jerome (1990), Model Theory, Studies in Logic and the Foundations of Mathematics (3rd ed.), Elsevier, ISBN 978-0-444-88054-3
• Hodges, Wilfrid (1997), A shorter model theory, Cambridge University Press, ISBN 978-0-521-58713-6