An improper arithmetically closed Borel subalgebra of [formula omitted] mod FIN

Abstract

We show the existence of a subalgebra A ⊆ P ( ω ) that satisfies the following three conditions: • A is Borel (when P ( ω ) is identified with 2 ω ). • A is arithmetically closed (i.e., A is closed under the Turing jump, and Turing reducibility). • The forcing notion ( A , ⊆ ) modulo the ideal FIN of finite sets collapses the continuum to ℵ 0 .