  Many extensions of first-order languages can be translated into infinitary languages. For example, consider the generalized quantifier language  L(Q0) obtained from  L  by adding a new quantifier symbol Q0 and interpreting Q0xφ(x) as there exist infinitely many x such that φ(x). It is easily seen that the sentence Q0xφ(x) has the same models as the  L(ω1,ω)-sentence