We suggest the name “approximation theorems” for a new kind of theorems which are strong versions of preservation theorems. A typical preservation theorem has the form:
A sentence φ is preserved by the relation R (i.e. and imply ) iff there exists φ* ϵ Φ such that ⊧φ↔ φ*
where R is a relation between structures and Φ is a class of sentences (depending, of course, on R). Usually, Φ is described in syntactical terms and it is easy to see that every element of it is, indeed, preserved by R. A typical approximation theorem has the form:
For every sentence Φ there is a sentence Φ* ϵ Φ (the “approximation” of Φ) such that, for all sentences δ which are preserved under R,
(a) if ⊧δ → φ then ⊧δ → φ* and
(b) if ⊧φ → δ then ⊧φ* → δ.
An approximation theorem obviously implies the corresponding preservation theorem.
The first approximation theorem was proved by Vaught in  (see Corollary 2.3 below). That paper inspired the present one. Vaught's result is stated in topological terms. It says that for each Borel set B there is an invariant Borel set B* explicitly defined by an Lω1ω sentence, such that B− ⊆ B* ⊆ B+ where B− (B+) is the largest (smallest) invariant set included in (containing) B.