Bar recursion in classical realisability : dependent choice and continuum hypothesis

TL;DR

This paper demonstrates that classical realizability models with bar recursion satisfy the axioms of dependent choice, well-ordering of reals, and the continuum hypothesis, enabling program extraction from proofs involving these axioms.

Contribution

It provides a new proof using classical realizability that models of set theory with bar recursion satisfy key axioms like dependent choice and continuum hypothesis.

Findings

Realizability models satisfy the axiom of dependent choice.

Models also satisfy the well-ordering of the real numbers.

The continuum hypothesis is realized by closed λ_c-terms.

Abstract

This paper is about the bar recursion operator in the context of classical realizability. After the pioneering work of Berardi, Bezem & Coquand [1], T. Streicher has shown [10], by means of their bar recursion operator, that the realizability models of ZF, obtained from usual models of $\lambda$-calculus (Scott domains, coherent spaces, . . .), satisfy the axiom of dependent choice. We give a proof of this result, using the tools of classical realizability. Moreover, we show that these realizability models satisfy the well ordering of $\mathbb{R}$ and the continuum hypothesis These formulas are therefore realized by closed $\lambda_c$-terms. This allows to obtain programs from proofs of arithmetical formulas using all these axioms.