Adding $\aleph_ω$ many Cohen reals
Pedro Marun, Saharon Shelah, Corey Bacal Switzer
TL;DR
This work investigates a concrete combinatorial principle, $\mathsf{Pr}(\sigma,\theta,\mu,\lambda)$, to distinguish the generic extensions produced by adding different magnitudes of Cohen reals. The authors prove a forward direction: adding $\mu$ many Cohen reals (under suitable hypotheses, including $2^\sigma=\theta<\mu<\lambda$ and a $\sigma^+$-covering property) enforces $\mathsf{Pr}(\sigma,\theta,\mu,\lambda)$; in particular, under GCH, $\mathrm{Add}(\omega,\aleph_\omega)$ yields $\mathsf{Pr}(\aleph_0,\aleph_1,\aleph_\omega,\aleph_{\omega+1})$. The authors then show the opposite direction: adding $\mu^+$ many Cohen reals forces the negation of this principle for a strong-limit $\mu$ with $\operatorname{cf}(\mu)=\omega$. Together, these results establish a precise combinatorial boundary that distinguishes the two forcing models. The paper generalizes the main dichotomy to broader settings (e.g., $\mu$ strong limit of countable cofinality) and frames questions about the principle’s broader implications and connections to other structures. This work thus links forcing density properties of Boolean algebras to intrinsic combinatorial criteria distinguishing different Cohen-forcing extensions.
Abstract
Abstractly, the generic extensions after $\aleph_ω$-many Cohen reals and $\aleph_{ω+1}$-many Cohen reals must be different for reasons of uniform density the relevant Boolean algebras. Nevertheless this is not satisfying and it would be nice to pin the difference between the two models down to some mathematical or combinatorial principle. In this paper we provide such a principle.