model

This page is about the notion of *model* in logic. For the notion in physics see *model (in theoretical physics)*.

In model theory, a **model** of a theory is a realization of the types, operations, relations, and axioms of that theory. In ordinary model theory one usually studies mainly models in sets, but in categorical logic we study models in other categories, especially in topoi.

The term *structure* is often used to mean a realization of types, operations, and relations in some signature, but not satisfying any particular axioms. This is of course the same as a model for the “empty theory” in that signature, which has the same types, operations, and relations, but no axioms at all. One then talks about whether a given structure is, or is not, a *model* of a given theory in a given signature.

The basic concept is of a *structure* for a first-order language $L$: a set $M$ together with an interpretation of $L$ in $M$. A theory $T$ is specified by a language and a set of sentences in $L$.

An $L$-structure $M$ is a **model** of $T$ if for every sentence $\phi$ in $T$, its interpretation in $M$, $\phi^M$ is true (“$\phi$ holds in $M$”).

We say that $T$ is **consistent** or satisfiable (relative to the universe in which we do model theory) if there exist at least one model for $T$ (in our universe). Two theories, $T_1$, $T_2$ are said to be **equivalent** if they have the same models.

Given a class $K$ of structures for $L$, there is a theory $Th(K)$ consisting of all sentences in $L$ which hold in every structure from $K$. Two structures $M$ and $N$ are **elementary equivalent** (sometimes written by equality $M=N$, sometimes said “elementarily equivalent”) if $Th(M)=Th(N)$, i.e. if they satisfy the same sentences in $L$. Any set of sentences which is equivalent to $Th(K)$ is called a **set of axioms** of $K$. A theory is said to be **finitely axiomatizable** if there exist a finite set of axioms for $K$.

A theory is said to be **complete** if it is equivalent to $Th(M)$ for some structure $M$.

For $Syn(T)$ the syntactic category of an algebraic theory (where $Syn(T)$ is perhaps better known as a Lawvere theory), and for $C$ any category with finite limits, a *model* for $T$ in $C$ is a product-preserving functor

$N : Syn(T) \to C
\,.$

The **category of models** in this case is hence the full subcategory of the functor category $[Syn(T),C]$ on product-preserving functors.

Revised on February 16, 2015 20:13:23
by Todd Trimble
(67.81.95.215)