We prove a topological completeness theorem for the modal logic \(\textsf{GLP}\) containing operators \(\{\langle \xi \rangle :\xi \in \textsf{Ord}\}\) intended to capture a wellordered sequence of consistency operators increasing in strength. More specifically, we prove that, given a tall-enough scattered space X, any sentence \(\phi \) consistent with \(\textsf{GLP}\) can be satisfied on a polytopological space based on finitely many Icard topologies constructed over X and corresponding to the finitely many modalities that occur in \(\phi \).

中文翻译：

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超限可证明性逻辑的拓扑完备性定理

我们证明了包含运算符\(\{\langle \xi \rangle :\xi \in \textsf{Ord}\}\) 的模态逻辑 \(\textsf{ GLP } \) 的拓扑完备性定理，旨在捕获有序一致性运算符的序列强度增加。更具体地说，我们证明，给定一个足够高的分散空间X，任何符合\( \textsf{GLP}\) 的句子 \(\phi \)都可以在基于有限多个 Icard 拓扑构造的多拓扑空间上得到满足X并对应于\(\phi \)中出现的有限多模态。