Atomic model

In model theory , a branch of mathematical logic, an atomic model is a model that only realizes very few types and is therefore very small in a certain sense. The term is related to the term of the prime model and is dual to the term of the saturated model, which in turn realizes very many types.

definition

In the following, the languages ​​considered are always at most countable.

• A type above a theory is an atomic type if there is a formula such that it holds for all : ${\displaystyle t}$ ${\displaystyle {\mathcal {T}}}$ ${\displaystyle \psi ({\vec {x}})}$ ${\displaystyle \phi ({\vec {x}})\in t}$

${\displaystyle T\vdash \psi ({\vec {x}})\rightarrow \phi ({\vec {x}})}$

An atomic type is also called an isolated type or main type.

• A model of a theory is atomic if it only realizes atomic types, i.e. if for each type: is atomic over . ${\displaystyle {\mathfrak {A}}}$ ${\displaystyle {\mathcal {T}}}$ ${\displaystyle {\vec {a}}\in A^{n}}$ ${\displaystyle \{\phi ({\vec {x}})|{\mathfrak {A}}\vDash \phi ({\vec {a}})\}}$ ${\displaystyle {\mathcal {T}}}$

Since finite subsets of types are always realized, atomic (isolated) types are always realized. For countable languages ​​the Omitting-Types-Theorem says that these are exactly the types that are always realized.

Examples

• The theory of the real closed fields has the atomic model of the real algebraic numbers .
• Every finite model is atomic.
• Every model of empty language is atomic. In particular, this example provides uncountable atomic models.
• Any model of dense linear order theory with no endpoints is atomic.

existence

The following sentence applies:

• A complete theory has an atomic model if and only if the isolated (atomic) types are close to it.

characterization

• Countable atomic models of a complete theory are isomorphic.
• A model of a complete theory is a countable atomic model if and only if it is a prime model. ${\displaystyle {\mathcal {T}}}$

However, in many theories there are uncountable atomic models, which are therefore not prime models.

• If a complete theory has a countable saturated model, it also has an atomic model.

The principle cannot be reversed: The theory of the real closed bodies has an atomic model, but not a saturated model.

Never two

An application of the theory of atomic and saturated models is the following theorem proved by Vaught :

• No complete theory (about a countable language) has exactly two countable non-isomorphic models. ${\displaystyle {\mathcal {T}}}$

In the proof one uses the fact that with exactly two non-isomorphic models one would have to be atomic ( ) and the other ( ) saturated. Now , if a non-isolated type is realized , one considers the theory of . With the theory of atomic and saturated models one can then conclude that there must be an atomic model of this theory and that the reduced model of this theory can neither be isomorphic to nor to be. ${\displaystyle {\mathfrak {A}}}$ ${\displaystyle {\mathfrak {B}}}$ ${\displaystyle b\in B}$ ${\displaystyle ({\mathfrak {B}},b)}$ ${\displaystyle {\mathcal {T}}}$ ${\displaystyle {\mathfrak {A}}}$ ${\displaystyle {\mathfrak {B}}}$

For example, the theory of language has exactly three countable models. is the atomic model. The saturated model is the model . The model between these two is . (The constant symbols are always interpreted by the natural numbers of the first copy of ). In the last model, the sequence of constants has a supremum, it is neither atomic nor saturated. ${\displaystyle \mathbb {Q} }$ ${\displaystyle \{<\}\cup \{n|n\in \mathbb {N} \}}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle (\mathbb {Q} +\mathbb {Q} )}$ ${\displaystyle (\mathbb {Q} +\mathbb {Q} _{0}^{+})}$ ${\displaystyle \mathbb {Q} }$

The theory can be extended by adding one- digit predicate symbols so that it has exactly models: The axioms are extended by the statements that exactly one predicate applies to each element, that the constants satisfy the predicate and that they are close together. Then there is again an atomic and a saturated model. In addition, there are models in which the sequence of constants has a supremum, but each of which fulfills a different predicate. ${\displaystyle n-2}$ ${\displaystyle P_{i}}$ ${\displaystyle n}$ ${\displaystyle P_{0}}$ ${\displaystyle \{x|P_{i}(x)\}}$ ${\displaystyle n-2}$

literature

• Wilfrid Hodges : Model theory. Cambridge University Press, 1993, ISBN 0-521-30442-3 .
• Chang, Chen C., Keisler, H. Jerome: Model theory. Third edition. Studies in Logic and the Foundations of Mathematics, 73. North-Holland Publishing Co., Amsterdam, 1990. ISBN 0-444-88054-2
• Prestel, Alexander: Introduction to Mathematical Logic and Model Theory. Vieweg, Braunschweig 1986. (Vieweg course; 60: advanced course in mathematics). ISBN 3-528-07260-1 .
• Philipp Rothmaler: Introduction to Model Theory. Spektrum Akademischer Verlag, 1995, ISBN 978-3-86025-461-5 .

Web links

• Martin Ziegler: Script Model Theory 1 . (PDF; 649 kB)

typography

1. ↑ When used in the following , it is an abbreviation for . So is z. B. ${\displaystyle {\vec {x}}}$ ${\displaystyle x_{1},\dots x_{n}}$

   ${\displaystyle \phi ({\vec {x}})}$

   an abbreviation of

   ${\displaystyle \phi (x_{1},\dots ,x_{n})}$.