(Classical) Model Theory

As it became clear at the end of the last section a formal semantic presupposes an interpretation function $\cal{I}$ mapping some language $\cal{S}$ to some abstract structure $\cal{A}$. Traditionally this has been called Model Theory. I would like to call it Classical Model Theory because the usual application context of this kind of theory was the language of first order logic (predicate logic) $\cal{S}$ and the construction of a model for this language. Introducing the logical inference concept $\vdash$ in parallel with the logical satisfaction concept $\models$ yields the following conceptual symmetry (cf. Chang 1973 [52]:33) for some set of expressions $\Sigma \subseteq \cal{S}$ :

SYNTAX	SEMANTICS
$\vdash \phi$ ($\phi$ is a theorem)	$\models \phi$ ($\phi$ is valid)
$\Sigma$ is consistent	$\Sigma$ has a model
$\Sigma \vdash \phi$ ($\phi$ is deducible from $\Sigma$)	$\Sigma \models \phi$ ($\phi$ is a consequence from $\Sigma$)

Assuming $\cal{A}$ as a model and $\sigma$ as a sentence of the language $\cal{S}$, then it is common to circumscribe the expression

$\cal{A} \models \sigma$ as follows:

$\sigma$ holds in $\cal{A}$

$\cal{A}$ satisfies $\sigma$

$\sigma$ is satisfied in $\cal{A}$

$\cal{A}$ is a model of $\sigma$

Examples of theorems in (classical) model theory are the following ones :

Gödel's Completeness Theorem: Given any sentence $\sigma$, $\sigma$ is a theorem of $\cal{S}$ if and only if $\sigma$ is valid.

Extended Completeness Theorem: Let $\Sigma$ be any set of sentences, then $\Sigma$ is consistent if and only if $\Sigma$ has a model.

Compatness Theorem: A set of sentences $\Sigma$ has a model if and only if every finite subset of $\Sigma$ has a model.