On elementary equivalence for fragments of language \(L_{\omega_ 1,\omega}\) (Q1901896)

We study fragments of the language \(L_{\omega_1, \omega}\), which are described by means of one formula for any finitely generated model up to isomorphism. These fragments are called finitely categorical languages. The lack of the compactness theorem for certain finitely categorical languages, both in J. Barwise's form and in classical form, calls for a new technique for answering model-theoretic questions. Here we offer a technique permitting to describe equivalent models for finitely categorical languages. For logics of first order, Ehrenfeucht games give a suitable criterion for elementary equivalence. Naturally, there is a wish to generalize this method to the languages under consideration. We show that for finitely categorical languages the Ehrenfeucht criterion is not a necessary condition for equivalence. We find necessary and sufficient conditions for the validity of the Ehrenfeucht criterion.