Atomic model (mathematical logic)
In
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.[1] 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 orderingwithout endpoints is atomic.
- Any 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
isomorphic
.
Notes
- ^ 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
- ISBN 978-0-444-88054-3
- ISBN 978-0-521-58713-6
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