Namba forcing and the weak approximation property
Maxwell Levine
Research in infinitary combinatorics has shown that the specific cardinals \(\aleph_0\), \(\aleph_1\), \(\aleph_2\), etc\(.\) exhibit distinct properties. One way of studying these distinctions is to examine to what extent these cardinals can be turned into one another by forcing. Bukovsky and Namba independently showed that \(\aleph_2\) can be turned into an ordinal of cofinality \(\aleph_0\) without collapsing \(\aleph_1\), and this forcing and its variants for other cardinals are now known as Namba forcing. In this talk we will show that it is consistent from an inaccessible cardinal that classical Namba forcing has the weak \(\omega_1\)-approximation property, answering a question of Cox and Krueger. The exact statement we obtain is similar to Hamkins’ Key Lemma and has implications for weakly guessing models. Time permitting, we will discuss implications for the study of successors of singular cardinals like \(\aleph_{\omega+1}\).