A simple computational interpretation of set theory

TL;DR

This paper presents a simplified constructive interpretation of set theory, modeling CZF using computable, well-founded set enumerations, and explores its implications for foundational principles and choice axioms.

Contribution

It introduces a new, intuitive family of constructive interpretations of CZF based on computable set enumerations, simplifying previous models.

Findings

Models CZF using well-founded computable set enumerations.

Provides a conservativity result for certain choice and Omniscience principles.

Models the Regular Extension Axiom within this framework.

Abstract

CZF is a system of set theory which, over classical logic, is equivalent to ZF, while over intuitionistic logic, it has a well-known constructive type-theoretic interpretation. This article introduces a simpler, intuitive family of constructive interpretations: sets are well-founded extensional computable conditional enumerations of sets. One interpretation in this family is just this: all sets are inductively built from the empty set by iterating the construction {f_n | n:N ^ g_n = h_n}, where, in turn, g and h are computable sequences of sets, and f is a computable sequence such that f_n is a set when g_n and h_n are extensionally equal. Extended Church's Thesis, an assumption which is incompatible with classical logic, is required to make this a model of CZF. Besides its foundational interest, it yields a direct conservativity result for certain choice principles, the Subcountability axiom, and for some so-called Omniscience principles, including first-order arithmetic Omniscience. A larger interpretation in this family also models the Regular Extension Axiom.