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