This paper contributes to the generalization of lattice-valued models of set theory to non-classical contexts. First, we show that there are infinitely many complete bounded distributive lattices, which are neither Boolean nor Heyting algebra, but are able to validate the negation-free fragment of $$\mathsf {ZF}$$ ZF . Then, we build lattice-valued models of full $$\mathsf {ZF}$$ ZF , whose internal logic is weaker than intuitionistic logic. We conclude by using these models to give an independence proof of the Foundation axiom from $$\mathsf {ZF}$$ ZF .

中文翻译：

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ZF 的非经典模型

本文有助于将集合论的格值模型推广到非经典环境。首先，我们证明存在无限多个完全有界分配格，它们既不是布尔代数也不是 Heyting 代数，但能够验证 $$\mathsf {ZF}$$ ZF 的无否定片段。然后，我们建立完整的 $$\mathsf {ZF}$$ ZF 的格值模型，其内部逻辑比直觉逻辑弱。最后，我们使用这些模型从 $$\mathsf {ZF}$$ ZF 给出基础公理的独立性证明。