On Namba Forcing and Minimal Collapses

TL;DR

This paper extends previous work on Namba forcing by removing the continuum hypothesis assumption and demonstrates the existence of specific minimal extensions using a measurable cardinal, answering longstanding questions.

Contribution

It removes the CH hypothesis from a minimality result and constructs a minimal extension with a measurable cardinal that differs at higher cardinals.

Findings

Removed CH hypothesis from a minimality result.

Constructed a minimal extension using a measurable cardinal.

Answered a question about the existence of certain minimal extensions.

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.