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.

## Contents

* 1 Definitions
* 2 Examples
* 3 Properties
* 4 Notes
* 5 References

## 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 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

1. ↑ 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

* Chang, Chen Chung; Keisler, H. Jerome (1990), Model Theory, Studies in Logic and the Foundations of Mathematics (3rd ed.), Elsevier, ISBN 978-0-444-88054-3
* Hodges, Wilfrid (1997), A shorter model theory, Cambridge University Press, ISBN 978-0-521-58713-6

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Mathematical logic

General|

* Formal language
* Formation rule
* Formal proof
* Formal semantics
* Well-formed formula
* Set
* Element
* Class
* Classical logic
* Axiom
* Rule of inference
* Relation
* Theorem
* Logical consequence
* Type theory
* Symbol
* Syntax
* Theory

Systems|

* Formal system
* Deductive system
* Axiomatic system
* Hilbert style systems
* Natural deduction
* Sequent calculus

Traditional logic|

* Proposition
* Inference
* Argument
* Validity
* Cogency
* Syllogism
* Square of opposition
* Venn diagram

Propositional calculus
and Boolean logic

* Boolean functions
* Propositional calculus
* Propositional formula
* Logical connectives
* Truth tables
* Many-valued logic

Predicate logic|

* First-order
* Quantifiers
* Predicate
* Second-order
* Monadic predicate calculus

Naive set theory|

* Set
* Empty set
* Element
* Enumeration
* Extensionality
* Finite set
* Infinite set
* Subset
* Power set
* Countable set
* Uncountable set
* Recursive set
* Domain
* Codomain
* Image
* Map
* Function
* Relation
* Ordered pair

Set theory|

* Foundations of mathematics
* Zermelo–Fraenkel set theory
* Axiom of choice
* General set theory
* Kripke–Platek set theory
* Von Neumann–Bernays–Gödel set theory
* Morse–Kelley set theory
* Tarski–Grothendieck set theory

Model theory|

* Model
* Interpretation
* Non-standard model
* Finite model theory
* Truth value
* Validity

Proof theory|

* Formal proof
* Deductive system
* Formal system
* Theorem
* Logical consequence
* Rule of inference
* Syntax

Computability
theory|

* Recursion
* Recursive set
* Recursively enumerable set
* Decision problem
* Church–Turing thesis
* Computable function
* Primitive recursive function

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