Cauchy completeness in elementary logic

Abstract

AbstractThe inverse of the distance between two structures≢of finite typeτis naturally measured by the smallest integerqsuch that a sentence of quantifier rankq− 1 is satisfied bybut not by. In this way the space Strτof structures of typeτis equipped with a pseudometric. The induced topology coincides with the elementary topology of Strτ. Using the rudiments of the theory of uniform spaces, in this elementary note we prove the convergence of every Cauchy net of structures, for any typeτ.