CAUCHY COMPLETENESS IN ELEMENTARY LOGIC

The inverse of the distance between two structures A not equal B of fi nite type sis naturally measured by the smallest integer q such that a sentence of quantifier rank q-1 is satisfied by A but not by B. In th is way the space Str of structures of type tau is equipped with a pseu dometric. The induced topology coincides with the elementary topology of Str(tau). Using the rudiments of the theory of uniform spaces, in t his elementary note we prove the convergence of every Cauchy net of st ructures, for any type tau.