Given a regular cardinal $\kappa $ such that $\kappa ^{<\kappa }=\kappa $ (or any regular $\kappa $ if the Generalized Continuum Hypothesis holds), we study a class of toposes with enough points, the $\kappa $ -separable toposes. These are equivalent to sheaf toposes over a site with $\kappa $ -small limits that has at most $\kappa $ many objects and morphisms, the (basis for the) topology being generated by at most $\kappa $ many covering families, and that satisfy a further exactness property T. We prove that these toposes have enough $\kappa $ -points, that is, points whose inverse image preserve all $\kappa $ -small limits. This generalizes the separable toposes of Makkai and Reyes, that are a particular case when $\kappa =\omega $ , when property T is trivially satisfied. This result is essentially a completeness theorem for a certain infinitary logic that we call $\kappa $ -geometric, where conjunctions of less than $\kappa $ formulas and existential quantification on less than $\kappa $ many variables is allowed. We prove that $\kappa $ -geometric theories have a $\kappa $ -classifying topos having property T, the universal property being that models of the theory in a Grothendieck topos with property T correspond to $\kappa $ -geometric morphisms (geometric morphisms the inverse image of which preserves all $\kappa $ -small limits) into that topos. Moreover, we prove that $\kappa $ -separable toposes occur as the $\kappa $ -classifying toposes of $\kappa $ -geometric theories of at most $\kappa $ many axioms in canonical form, and that every such $\kappa $ -classifying topos is $\kappa $ -separable. Finally, we consider the case when $\kappa $ is weakly compact and study the $\kappa $ -classifying topos of a $\kappa $ -coherent theory (with at most $\kappa $ many axioms), that is, a theory where only disjunction of less than $\kappa $ formulas are allowed, obtaining a version of Deligne’s theorem for $\kappa $ -coherent toposes from which we can derive, among other things, Karp’s completeness theorem for infinitary classical logic.

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德利涅完全性定理的无限推广

给定一个普通的红衣主教$\卡帕$这样$\kappa ^{<\kappa }=\kappa $（或任何常规$\卡帕$如果广义连续统假设成立），我们研究一类具有足够点的拓扑，$\卡帕$- 可分离的姿势。这些相当于在一个站点上的捆拓扑$\卡帕$- 最多有的小限制$\卡帕$许多对象和态射，拓扑的（基础）最多由$\卡帕$许多覆盖家庭，并且满足进一步的精确性吨. 我们证明这些拓扑足够$\卡帕$-points，即逆像保留所有点的点$\卡帕$-小限制。这概括了 Makkai 和 Reyes 的可分离拓扑，当$\kappa =\omega $, 当属性吨是微不足道的满足。这个结果本质上是一个我们称之为无限逻辑的完备性定理$\卡帕$-几何，其中小于的连词$\卡帕$小于的公式和存在量化$\卡帕$允许许多变量。我们证明$\卡帕$-几何理论有一个$\卡帕$-对具有属性的拓扑进行分类吨, 普遍性质是具有性质的格洛腾迪克拓扑中的理论模型吨相当于$\卡帕$-几何态射（几何态射，其逆像保留所有$\卡帕$-小限制）进入该拓扑。此外，我们证明$\卡帕$- 可分离的toposes发生为$\卡帕$- 分类拓扑$\卡帕$- 至多的几何理论$\卡帕$规范形式的许多公理，并且每一个这样的公理$\卡帕$-分类拓扑是$\卡帕$-可分离。最后，我们考虑以下情况$\卡帕$是弱紧致并研究$\卡帕$- 对 a 的拓扑进行分类$\卡帕$- 连贯理论（至多$\卡帕$许多公理），即只有小于的析取的理论$\卡帕$允许使用公式，获得 Deligne 定理的一个版本$\卡帕$-我们可以从中推导出卡普关于无限经典逻辑的完备性定理的连贯拓扑。