On the strength of the finite intersection principle

TL;DR

This paper analyzes the logical strength of the finite intersection principle ($F\IP$) in set theory, revealing its varied strength in reverse mathematics and its relation to other principles like $\ACA$, $\WKL$, and $\AMT$.

Contribution

It establishes the reverse mathematical strength of $F\IP$, showing its equivalences, non-equivalences, and implications with other logical principles.

Findings

Existence of a computable $F\IP$ instance with solutions of hyperimmune degree

Every computable $F\IP$ instance has solutions in every nonzero c.e. degree

$F\IP$ implies the omitting partial types principle ($\mathsf{OPT}$)

Abstract

We study the logical content of several maximality principles related to the finite intersection principle ($F\IP$) in set theory. Classically, these are all equivalent to the axiom of choice, but in the context of reverse mathematics their strengths vary: some are equivalent to $\ACA$ over $\RCA$, while others are strictly weaker, and incomparable with $\WKL$. We show that there is a computable instance of $F\IP$ all of whose solutions have hyperimmune degree, and that every computable instance has a solution in every nonzero c.e.\ degree. In terms of other weak principles previously studied in the literature, the former result translates to $F\IP$ implying the omitting partial types principle ($\mathsf{OPT}$). We also show that, modulo $\Sigma^0_2$ induction, $F\IP$ lies strictly below the atomic model theorem ($\mathsf{AMT}$).