$\mathsf{RCA}_0$ Proves the Collapse of the Intersection Principles Hierarchy

Abstract

This paper shows that the pairwise intersection principle ($2\mathrm{IP}$) implies, over $\mathsf{RCA}_0$, the existence of a Cohen $1$-generic real relative to every given set. Together with the known implications, this yields the equivalence over $\mathsf{RCA}_0$ of the finite intersection principle (FIP), $2\mathrm{IP}$, and $1$-GEN. In fact, FIP is equivalent to $n\mathrm{IP}$ for every fixed $n\geq 2$. These results give affirmative answers to Questions~1 and~2 of Cholak, Downey, and Igusa, and to Question~9.10.26 of Dzhafarov and Mummert. The proof uses cumulative, nested supports satisfying a finite Helly property and a decoding argument based on a least selected label. Neither $Σ^0_2$ induction nor $Σ^0_2$ bounding is required.