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 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.