The celebrated van Benthem characterization theorem states that on Kripke structures modal logic is the bisimulation-invariant fragment of first-order logic. In this paper, we prove an analogue of the van Benthem characterization theorem for models based on descriptive general frames. This is an important class of general frames for which every modal logic is complete. These frames can be represented as Stone spaces equipped with a ‘continuous’ binary relation. The proof of our theorem generalizes Rosen’s proof of the van Benthem theorem for finite frames and uses as an essential technique a new notion of descriptive unravelling. We also develop a basic model theory for descriptive general frames and show that in many ways it behaves like the model theory of finite structures. In particular, we prove the failure of the compactness theorem, of the Beth definability theorem, of the Craig interpolation theorem and of the upward Löwenheim–Skolem theorem.¹

中文翻译：

---

描述通用框架的模型理论方法：van Benthem表征定理

著名的van Benthem刻画定理指出，在Kripke结构上，模态逻辑是一阶逻辑的双仿真不变片段。在本文中，我们证明了基于描述性通用框架的模型的van Benthem表征定理的类似物。这是一类重要的通用框架，其所有模态逻辑均已完成。这些框架可以表示为具有“连续”二进制关系的Stone空间。我们的定理的证明推广了罗森（Rosen）对范本思定理的有限框架的证明，并将描​​述性拆散的新概念用作基本技术。我们还开发了用于描述性通用框架的基本模型理论，并显示了它在许多方面的行为类似于有限结构的模型理论。特别是，我们证明了紧性定理的失败，^1个