Atomic model

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In model theory, an atomic model is a model which is in some sense small.

Contents 1 Definitions 2 Examples 3 Properties 4 References

[edit] Definitions

A formula φ(x₁,...,x_n) 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.

A model M of the theory is called atomic if every n-tuple of elements of M satisfies a complete formula.

[edit] 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.
• 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.

[edit] Properties

The back and forth method can be usesd to show that any two countable atomic models of a theory that are elementarily equivalent are isomorphic.

[edit] References

Chang, Chen Chung & Keisler, H. Jerome (1990), Model Theory (3rd ed.), Studies in Logic and the Foundations of Mathematics, Elsevier, ISBN 978-0-444-88054-3