Journal of Symbolic Logic

Cauchy Completeness in Elementary Logic

J. C. Cifuentes, A. M. Sette, and D. Mundici

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The inverse of the distance between two structures $\mathscr{A} \not\equiv \mathscr{B}$ of finite type $\tau$ is naturally measured by the smallest integer $q$ such that a sentence of quantifier rank $q - 1$ is satisfied by $\mathscr{A}$ but not by $\mathscr{B}$. In this way the space $\operatorname{Str}^\tau$ of structures of type $\tau$ 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 $\tau$.

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J. Symbolic Logic, Volume 61, Issue 4 (1996), 1153-1157.

First available in Project Euclid: 6 July 2007

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Cifuentes, J. C.; Sette, A. M.; Mundici, D. Cauchy Completeness in Elementary Logic. J. Symbolic Logic 61 (1996), no. 4, 1153--1157.

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