nLab
model

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

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Idea

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.

Definition

In first-order theories

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 $\varphi$ in $T$, its interpretation in $M$, ${\varphi }^M$ is true (”$\varphi$ 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 $\mathrm{Th}(K)$ 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 $\mathrm{Th}(M)=\mathrm{Th}(N)$, i.e. if they satisfy the same sentences in $L$. Any set of sentences which is equivalent to $\mathrm{Th}(K)$ 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 $\mathrm{Th}(M)$ for some structure $M$.

For Lawvere theories

For $\mathrm{Syn}(T)$ the syntactic category of a Lawvere theory, and for $C$ any category with finite limits, a model for $T$ in $C$ is a product-preserving functor

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

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

Related entries

• structure in model theory