In mathematics, specifically model theory, a class K of models for a first-order language L is an elementary class if there is some sentence τ in the language such that for all models A, iff A satisfies τ.

Examples

Let L be a language consisting only of a unary operator F. Let K be the class of models in which F is one-to-one. K is an elementary class, since we can let

τ = "".

Since this is the very definition of being one-to-one, this certainly does the job.

On the other hand, let L be any language. The class of all infinite models is not elementary. Suppose towards contradiction that there is some sentence that does the job. So, says exactly "there are only a finite number of objects in the universe". Let

ρ₂="".

Let

ρ₃="".

For , define ρ_n similarly; that is, to say "there are at least n objects in the universe". Let

.

For every finite , B is clearly satisfied in some model. However, there is no model that satisfies A in its entirety, which violates the compactness theorem. So, the class of all infinite models is not elementary. This argument also yields the result that the class of all finite models is not elementary.

The fore-mentioned does exist in second-order logic, which means that the Compactness Theorem does not hold there.