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arXiv · 2411.19907

Choice and independence of premise rules in intuitionistic set theory

Emanuele Frittaion·Takako Nemoto·Michael Rathjen
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Abstract

Choice and independence of premise principles play an important role in characterizing Kreisel's modified realizability and G\"odel's Dialectica interpretation. In this paper we show that a great many intuitionistic set theories are closed under the corresponding rules for finite types over $\mathbb{N}$. It is also shown that the existence property (or existential definability property) holds for statements of the form $\exists y^{\sigma}\, \varphi(y)$, where the variable $y$ ranges over objects of finite type $\sigma$. This applies in particular to ${\sf CZF}$ (Constructive Zermelo-Fraenkel set theory) and ${\sf IZF}$ (Intuitionistic Zermelo-Fraenkel set theory), two systems known not to have the general existence property. On the technical side, the paper uses a method that amalgamates generic realizability for set theory with truth, whereby the underlying partial combinatory algebra is required to contain all objects of finite type.

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Emanuele Frittaion, Takako Nemoto, Michael Rathjen. 2024-11-29. Choice and independence of premise rules in intuitionistic set theory. https://arxiv.org/abs/2411.19907

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