Prime model

In the model theory , a branch of mathematical logic , is called a model of a theory of prime model if this model in any model of this theory elementary can be embedded.

definition

The following is a countable theory with no finite models. ${\displaystyle {\mathcal {T}}}$

${\displaystyle {\mathfrak {A}}}$is a prime model of the theory if and only if for all with a picture exists with ${\displaystyle {\mathcal {T}}}$ ${\displaystyle {\mathfrak {B}}}$ ${\displaystyle {\mathfrak {B}}\models {\mathcal {T}}}$ ${\displaystyle \Phi }$ ${\displaystyle \Phi :{\mathfrak {A\hookrightarrow B}}}$

Examples

• The algebraic closure of the prime field (or ) is a prime model of the theory of the algebraically closed fields of the characteristic (or 0). ${\displaystyle \mathbb {F} _{p}}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle p}$
• ${\displaystyle \mathbb {Q} }$is a prime model of the dense linear orders without extrema .

properties

• From the Löwenheim-Skolem theorem it follows that a prime model is countable.
• If -categorical, then the countable model is a prime model. ${\displaystyle {\mathcal {T}}}$ ${\displaystyle \aleph _{0}}$
• Two prime models of a theory are isomorphic.
• A theory has a prime model if and only if the isolated types are close together.

Example of a theory without a prime model

The following theory of language has no prime model: Language contains a single-digit predicate for each . ${\displaystyle L}$ ${\displaystyle L}$ ${\displaystyle s\in }$ ${\displaystyle {}^{<\omega }2}$ ${\displaystyle P_{s}}$

(On the notation: is the set of all finite sequences that consist only of zeros or ones.) ${\displaystyle {}^{<\omega }2}$

The axioms of the theory are ( runs through all finite sequences): ${\displaystyle s}$

${\displaystyle \forall xP_{0}(x)}$

${\displaystyle \exists xP_{s}(x)}$

${\displaystyle \forall x((P_{s0}(x)\lor P_{s1}(x))\leftrightarrow P_{s}(x))}$

${\displaystyle \forall x\neg (P_{s0}(x)\land P_{s1}(x))}$

The theory has no isolated types and therefore no prime model.

literature

• Wilfrid Hodges : Model theory. Cambridge University Press, 1993, ISBN 0-521-30442-3 .
• Chang, Chen C., Keisler, H. Jerome: Model Theory. Amsterdam [u. a.], North-Holland, 1998.
• Prestel, Alexander: Introduction to Mathematical Logic and Model Theory. Vieweg, Braunschweig 1986. (Vieweg course; 60: advanced course in mathematics). ISBN 3-528-07260-1 . 286 pp.
• 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)