Atomic model (mathematical logic)
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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(x1, ..., xn) is called principal or atomic (relative to T) if it is axiomatized relative to T by a single formula φ(x1, ..., xn) ∈ p(x1, ..., xn).
A formula φ is called complete in T if for every formula ψ(x1, ..., xn), the theory T ∪ {φ} entails exactly one of ψ and ¬ψ.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.
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
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References
- {{Citation | last1=Chang | first1=Chen Chung |author1-link=Chen Chung Chang| last2=Keisler | first2=H. Jerome | author2-link=Howard Jerome Keisler | title=Model Theory | publisher=Elsevier | edition=3rd | series=Studies in Logic and the Foundations of Mathematics | isbn=978-0-444-88054-3 | year=1990}}
- {{Citation | last1=Hodges | first1=Wilfrid | author1-link=Wilfrid Hodges | title=A shorter model theory | publisher=Cambridge University Press | isbn=978-0-521-58713-6 | year=1997}}
{{Mathematical logic}}