## Journal of Symbolic Logic

### Cauchy Completeness in Elementary Logic

#### Abstract

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 $\tau$.

#### Article information

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

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183745128

Mathematical Reviews number (MathSciNet)
MR1456100

Zentralblatt MATH identifier
0871.03020

JSTOR