On Namba Forcing and Minimal Collapses

Abstract

We build on a 1990 paper of Bukovsky and Coplakova-Hartova. First, we remove the hypothesis of $\text{CH}$ 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.