AI 中文总结
研究几何逻辑可数片段（σ-相干逻辑）中可数σ-相干理论完备性，在命题和谓词逻辑下分别确定模型的拟波兰空间和范畴，公式解释为连续函数与函子，还通过研究“清醒性”概念扩展类比。
AI 中文摘要
我们给出了命题逻辑和谓词逻辑中可数σ-相干理论完备性的自包含证明。通过σ-相干逻辑，我们指的是几何逻辑中仅允许可数签名和可数无限析取的片段。在命题情形下，每个理论确定一个模型的拟波兰空间，公式（直至可证等价）被解释为从模型空间到Sierpinski空间的连续函数。在谓词情形下，每个理论确定一个模型的拟波兰范畴，公式（直至可证等价）被解释为从模型范畴到明显离散拟波兰空间的拟波兰范畴的连续函子。我们通过研究某些拓扑范畴的“清醒性”概念进一步扩展了这种类比。
英文摘要
We give self contained proofs of the completeness of countable $σ$-coherent theories, for both propositional and predicate logic. By $σ$-coherent logic we mean the fragment of geometric logic that only allows countable signatures and countably infinite disjunctions. In the propositional case, each theory determines a quasi-Polish space of models, and formulas (up to provable equivalence) are interpreted as continuous functions from the space of models to the Sierpinski space. In the predicate case, each theory determines a quasi-Polish category of models, and formulas (up to provable equivalence) are interpreted as continuous functors from the category of models to the quasi-Polish category of overt discrete quasi-Polish spaces. We further extend this analogy by investigating a notion of ``sobriety'' for certain topological categories.
