Abstract

We study relativizations and potential isomorphisms of quantale-valued approach spaces. We show how the category of generalized approach spaces and injective maps can be encoded using higher order structures, but under such an encoding generalized approach spaces need not have a relativization. We then provide an alternate encoding of generalized approach spaces using higher order structures which does always relativize.

We also consider potential isomorphisms between generalized approach spaces and give a necessary and sufficient condition for such a potential isomorphism to exist, which is absolute between models of set theory. We then consider an abstract notion of sentence for generalized approach spaces and prove a downward Löowenheim-Skolem theorem as well as a Tarski-Vaught theorem for such generalized sentences.

摘要

我们研究量化值方法空间的相对化和潜在同构。我们展示了如何使用高阶结构对广义进近空间和内射映射的类别进行编码，但是在这种编码下，广义进近空间无需相对化。然后，我们使用总是相对化的高阶结构提供一种通用方法空间的替代编码。

我们还考虑了广义逼近空间之间的潜在同构，并给出了存在这种潜在同构的必要和充分条件，这在集合论模型之间是绝对的。然后，我们考虑广义方法空间的句子的抽象概念，并证明此类广义句子的向下Löowenheim-Skolem定理以及Tarski-Vaught定理。