On a variant of Rado’s selection lemma and its equivalence with the Boolean prime ideal theorem

Abstract

We establish that, in ZF (i.e., Zermelo–Fraenkel set theory minus the Axiom of Choice AC), the statement

RLT: Given a set I and a non-empty set ${\mathcal{F}}$ of non-empty elementary closed subsets of 2^I satisfying the fip, if ${\mathcal{F}}$ has a choice function, then ${\bigcap\mathcal{F} \ne \emptyset}$,

which was introduced in Morillon (Arch Math Logic 51(7–8):739–749, 2012), is equivalent to the Boolean Prime Ideal Theorem (see Sect. 1 for terminology). The result provides, on one hand, an affirmative answer to Morillon’s corresponding question in Morillon (2012) and, on the other hand, a negative answer—in the setting of ZFA (i.e., ZF with the axiom of extensionality weakened to permit the existence of atoms)—to the question in Morillon (2012) of whether RLT is equivalent to Rado’s selection lemma.