We develop a number of variants of Lifschitz realizability for $\mathbf {CZF}$ by building topological models internally in certain realizability models. We use this to show some interesting metamathematical results about constructive set theory with variants of the lesser limited principle of omniscience including consistency with unique Church’s thesis, consistency with some Brouwerian principles and variants of the numerical existence property.

中文翻译：

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作为拓扑结构的 LIFSCHITZ 可实现性

我们开发了许多 Lifschitz 可实现性的变体，用于$\mathbf {CZF}$通过在某些可实现性模型中内部构建拓扑模型。我们用它来展示关于建设性集合论的一些有趣的元数学结果，这些结果具有较小有限的全知原则的变体，包括与独特的 Church 论点的一致性，与一些 Brouwerian 原则的一致性以及数值存在属性的变体。