R. E. Moore [Interval analysis (1966; Zbl 0176.133)] introduced a model \({\mathcal I}({\mathbb{R}})\) for the ”real numbers” whose elements are compact real intervals. In this paper the first-order theory of \({\mathcal I}({\mathbb{R}})\) is axiomatized and Tarski’s quantifier elimination method for real closed fields is modified to show that the elementary theory of \({\mathcal I}({\mathbb{R}})\) is decidable.