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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(x<sub>1</sub>,&nbsp;...,&nbsp;x<sub>n</sub>) is called principal or atomic (relative to T) if it is axiomatized relative to T by a single formula φ(x<sub>1</sub>,&nbsp;...,&nbsp;x<sub>n</sub>)&nbsp;∈&nbsp;p(x<sub>1</sub>,&nbsp;...,&nbsp;x<sub>n</sub>).

A formula φ is called complete in T if for every formula ψ(x<sub>1</sub>,&nbsp;...,&nbsp;x<sub>n</sub>), the theory T ∪ {φ} 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(M)—the theory of M.

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

References