Spectral function (model theory)

In model theory, a mathematical branch of logic, the spectral function of a cardinal number assigns the number of non-isomorphic models to a theory . The spectral problem for a theory is to find these values.

Definitions

Is a theory, then is the number of non-isomorphic models in that theory. is the class of all cardinal numbers. The function ${\displaystyle \mathrm {T} }$ ${\displaystyle \mathrm {I} (\lambda ,\mathrm {T} )}$ ${\displaystyle \mathrm {Cn} }$

${\displaystyle \mathrm {Cn} \to \mathrm {Cn} }$

${\displaystyle \lambda \mapsto \mathrm {I} (\lambda ,\mathrm {T} )}$

is called the spectral function. (This function is not a set, but a real class)

Examples

If the theory of algebraically closed fields (algebra) is a fixed characteristic , then ${\displaystyle \mathrm {T} }$

${\displaystyle \mathrm {I} (\omega ,\mathrm {T} )=\omega }$

and for is ${\displaystyle \lambda >\omega }$

${\displaystyle \mathrm {I} (\lambda ,\mathrm {T} )=1}$

Because the models are precisely described by their degree of transcendence. The countable models are precisely those with finite or countable degrees of transcendence, and for uncountable degrees of transcendence this already determines the cardinality of the body.

If the theory of above is language , then: ${\displaystyle \mathrm {T} }$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \{<\}\cup \{c_{q}|q\in \mathbb {Q} \}}$

${\displaystyle \mathrm {I} (2^{\omega },\mathrm {T} )=2^{2^{\omega }}}$

Every powerful subset of the irrational numbers determines a model of this theory. ${\displaystyle 2^{\omega }}$

properties

General means

${\displaystyle \mathrm {I} (\lambda ,\mathrm {T} )=1,}$

that the theory is categorical in this cardinal number .

The Löwenheim-Skolem theorem says for a theory with that ${\displaystyle \mathrm {T} }$ ${\displaystyle \alpha ,\beta \geq |\mathrm {T} |,}$

${\displaystyle \mathrm {I} (\alpha ,\mathrm {T} )\geq 1\Rightarrow \mathrm {I} (\beta ,\mathrm {T} )\geq 1}$

appraisal

With elementary considerations it can be shown that for a theory about a language and the following applies: ${\displaystyle \mathrm {L} }$ ${\displaystyle \lambda \geq |\mathrm {L} |}$

${\displaystyle \mathrm {I} (\lambda ,\mathrm {T} )\leq 2^{\lambda }}$

This estimate is the best possible, for certain and there is equality. ${\displaystyle \lambda }$ ${\displaystyle \mathrm {T} }$

literature

• Philipp Rothmaler: Introduction to Model Theory , Spektrum Akademischer Verlag 1995, ISBN 978-3-86025-461-5