On Namba Forcing and Minimal Collapses

Abstract

We build on a 1990 paper of Bukovsky and Coplakova-Hartova. First, we remove the hypothesis of $\textsf{CH}$ from one of their minimality results. Then, using a measurable cardinal, we show that there is a $|\aleph_2^V|=\aleph_1$-minimal extension that is not a $|\aleph_3^V|=\aleph_1$-extension, answering the first of their questions.