In generic realizability for set theories, realizers treat unbounded quantifiers generically. To this form of realizability, we add another layer of extensionality by requiring that realizers ought to act extensionally on realizers, giving rise to a realizability universe |$\mathrm{V_{ex}}(A)$| in which the axiom of choice in all finite types, |${\textsf{AC}}_{{\textsf{FT}}}$|⁠, is realized, where |$A$| stands for an arbitrary partial combinatory algebra. This construction furnishes ‘inner models’ of many set theories that additionally validate |${\textsf{AC}}_{{\textsf{FT}}}$|⁠, in particular it provides a self-validating semantics for |${\textsf{CZF}}$| (constructive Zermelo–Fraenkel set theory) and |${\textsf{IZF}}$| (intuitionistic Zermelo–Fraenkel set theory). One can also add large set axioms and many other principles.

中文翻译：

---

直觉集合论的可扩展可实现性

在集合理论的一般可实现性中，实现者一般都对待无穷量词。对于这种形式的可实现性，我们通过要求实现者应该对实现者进行扩展来增加可扩展性的另一层，从而产生了一个可实现性世界| $ \ mathrm {V_ {ex}}（A）$ | 其中实现了所有有限类型的选择公理| $ {\ textsf {AC}} _​​ {{\ textsf {FT}}} $ |⁠，其中| $ A $ | 代表任意的部分组合代数。这种构造提供了许多设置理论的“内部模型”，这些理论进一步验证了| $ {\ textsf {AC}} _​​ {{\ textsf {FT}}} $ |⁠，尤其是它为| $ { \ textsf {CZF}} $ | （建设性的Zermelo–Fraenkel集论）和| $ {\ textsf {IZF}} $ | （直觉的Zermelo–Fraenkel集论）。还可以添加大集公理和许多其他原理。