Cauchy completeness in elementary logic

Journal of Symbolic Logic 61 (4):1153-1157 (1996)
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The inverse of the distance between two structures $\mathscr{A} \not\equiv \mathscr{B}$ of finite type τ is 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 this way the space $\operatorname{Str}^\tau$ of structures of type τ is equipped with a pseudometric. The induced topology coincides with the elementary topology of $\operatorname{Str}^\tau$ . 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 type τ



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Models and Ultraproducts: An Introduction.J. L. Bell & A. B. Slomson - 1972 - Journal of Symbolic Logic 37 (4):763-764.
General Topology.John L. Kelley - 1962 - Journal of Symbolic Logic 27 (2):235-235.
Compactness and normality in abstract logics.Xavier Caicedo - 1993 - Annals of Pure and Applied Logic 59 (1):33-43.

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