Elementary class

Remarks

If $T$ can be chosen finite, or equivalently a single sentence, $K$ is called a basic elementary class. Some authors write $\mathrm{EC}$ for basic elementary classes and $\mathrm{EC}_\Delta$ for elementary classes.

Example: groups, in the signature $(\cdot, {}^{-1}, e)$, form a basic elementary class, axiomatized by the three group axioms. Fields of characteristic $0$ form an elementary class, using the infinitely many axioms $1 + \cdots + 1 \ne 0$; it is not a basic elementary class.

Non-example: torsion groups do not form an elementary class. Suppose $T$ axiomatized them. Add a new constant $c$ and the sentences $c^n \ne e$ for all $n \ge 1$. Each finite subset of this theory has a model, namely a finite cyclic group of large order with $c$ a generator. By the compactness theorem the whole theory has a model, which is a model of $T$ with an element of infinite order, a contradiction.