AI 中文总结
研究在一阶布尔教义形式体系内自证哥德尔完备性定理，表明该定理使一阶布尔教义的纤维式斯通对偶成为类型空间函子，阐述了公式布尔代数对偶与模型斯通空间的关系。
AI 中文摘要
我们在一阶布尔教义（经典多类一阶逻辑的代数方法）的形式体系内，给出了哥德尔完备性定理的自包含证明。此外，我们表明哥德尔完备性定理意味着一阶布尔教义的纤维式斯通对偶是其类型空间函子，大致来说，这意味着上下文\(X\)中公式的布尔代数的斯通对偶是\(X\)点模型模初等等价的斯通空间。
英文摘要
We give a self-contained proof of Gödel's completeness theorem entirely within the formalism of first-order Boolean doctrines (an algebraic approach to classical many-sorted first-order logic). Moreover, we show that Gödel's completeness theorem entails that the fiberwise Stone dual of a first-order Boolean doctrine is its type space functor; roughly speaking, this means that the Stone dual of the Boolean algebra of formulas in context $X$ is the Stone space of $X$-pointed models modulo elementary equivalence.
