Model completeness

In model theory , a sub-area of mathematical logic , a theory is called model-complete if sub-models are particularly good in its super-model.

definition

A theory is called model-complete if it holds for two models and of that it follows that elementary is in , in signs . ${\displaystyle {\mathcal {T}}}$ ${\displaystyle {\mathfrak {A}}}$ ${\displaystyle {\mathfrak {B}}}$ ${\displaystyle {\mathcal {T}}}$ ${\displaystyle {\mathfrak {B}}\subseteq {\mathfrak {A}}}$ ${\displaystyle {\mathfrak {B}}}$ ${\displaystyle {\mathfrak {A}}}$ ${\displaystyle {\mathfrak {B}}\prec {\mathfrak {A}}}$

Robinson's test

Robinson's test can often be used to demonstrate the completeness of the model. A formula of a language is called existential if it is of the form ${\displaystyle \phi }$ ${\displaystyle {\mathcal {L}}}$

${\displaystyle \exists x_{1},...,x_{n}\psi }$

with quantifier-free is. Analogously, a formula is called universal if it is of the form ${\displaystyle \psi }$

${\displaystyle \forall x_{1},...,x_{n}\psi }$

with quantifier-free is. If there are two models, then in is called existentially closed if every existential statement of the language that applies in also applies in. ${\displaystyle \psi }$ ${\displaystyle {\mathfrak {B\subseteq A}}}$ ${\displaystyle {\mathcal {L}}}$ ${\displaystyle {\mathfrak {B}}}$ ${\displaystyle {\mathfrak {A}}}$ ${\displaystyle {\mathcal {L}}_{\mathfrak {B}}}$ ${\displaystyle {\mathfrak {A}}}$ ${\displaystyle {\mathfrak {B}}}$

Robinson's test is:

For a set of propositions is equivalent: ${\displaystyle {\mathcal {T}}}$

1. ${\displaystyle {\mathcal {T}}}$ is model complete.
2. For two models of with is existentially closed in . ${\displaystyle {\mathfrak {B,A}}}$ ${\displaystyle {\mathcal {T}}}$ ${\displaystyle {\mathfrak {B}}\subseteq {\mathfrak {A}}}$ ${\displaystyle {\mathfrak {B}}}$ ${\displaystyle {\mathfrak {A}}}$
3. At any -formula there is a universal -formula whose free variables in the free variables of are included, then the equivalent of that and made without evidence. ${\displaystyle {\mathcal {L}}}$ ${\displaystyle \phi }$ ${\displaystyle {\mathcal {L}}}$ ${\displaystyle \psi }$ ${\displaystyle \phi }$ ${\displaystyle \phi }$ ${\displaystyle \psi }$ ${\displaystyle {\mathcal {T}}}$

Completeness versus model completeness

A complete theory does not have to be model-complete, nor does a model-complete theory have to be complete. But if a model-complete theory has a model that can be embedded in any other model of the theory, this theory is also complete. (see prime model )

Model companion

A theory is called a model companion of a theory , if ${\displaystyle {\mathcal {T}}^{*}}$ ${\displaystyle {\mathcal {T}}}$

• ${\displaystyle T\subseteq T^{*}}$
• each model can be expanded from to a model from and ${\displaystyle {\mathcal {T}}}$ ${\displaystyle T^{*}}$
• ${\displaystyle {\mathcal {T}}^{*}}$ is model-complete.

It can be shown that there is at most one model companion for each theory.

Examples

• The theory of dense linear open total order is complete and model-complete. It is a model companion of the theory of linear orders.
• The theory of algebraically closed bodies (without any statement about the characteristics ) is not complete, but it is model-complete.
• The theory of algebraically closed fields of a fixed characteristic has a prime model and is both complete and model-complete.
• The theory of dense linear total order with extrema is complete, but not model-complete. The interval is not fundamentally part of the interval . ${\displaystyle [0,1]}$ ${\displaystyle [0,2]}$

literature

• Chang, Chen C., Keisler, H. Jerome, Model Theory , Amsterdam [u. a.], North Holland (1998)
• Prestel: Introduction to Mathematical Logic and Model Theory, Braunschweig, Wiesbaden (1986)