We give a formalism for approximate isomorphism in continuous logic simultaneously generalizing those of two papers by Ben Yaacov [2] and by Ben Yaacov, Doucha, Nies, and Tsankov [6], which are largely incompatible. With this we explicitly exhibit Scott sentences for the perturbation systems of the former paper, such as the Banach-Mazur distance and the Lipschitz distance between metric spaces. Our formalism is simultaneously characterized syntactically by a mild generalization of perturbation systems and semantically by certain elementary classes of two-sorted structures that witness approximate isomorphism. As an application, we show that the theory of any $\mathbb{R}$-tree or ultrametric space of finite radius is stable, improving a result of Carlisle and Henson [8].

中文翻译：

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度量结构的近似同构

我们给出了连续逻辑中近似同构的形式主义，同时概括了 Ben Yaacov [2] 和 Ben Yaacov、Doucha、Nies 和 Tsankov [6] 的两篇论文，它们在很大程度上是不兼容的。这样，我们明确地展示了前一篇论文中扰动系统的斯科特句子，例如度量空间之间的 Banach-Mazur 距离和 Lipschitz 距离。我们的形式主义在句法上同时具有扰动系统的温和概括的特征，在语义上则具有见证近似同构的二分类结构的某些基本类的特征。作为一个应用，我们证明了任何理论$\mathbb{右}$- 有限半径的树或超度量空间是稳定的，改进了 Carlisle 和 Henson [8] 的结果。