There exists a d-minimal expansion of the $\mathbb{R}$-vector space over $\mathbb{R}$ which defines every sequence

Abstract

There exists a d-minimal expansion of the $\mathbb{R}$-vector space over $\mathbb{R}$ which defines every sequence. In this paper, we prove this assertion and the following more general assertion: Let $\mathcal{R}$ be either the ordered $\mathbb{R}$-vector space structure over $\mathbb{R}$ or the ordered group of reals. A first-order expansion of $\mathcal{R}$ by a countable subset $D$ of $\mathbb{R}$ and a compact subset $E$ of $\mathbb{R}$ of finite Cantor-Bendixson rank is d-minimal if $(\mathcal{R},D)$ is locally o-minimal.