Homogeneous Namba-like forcing - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T01:55:09Z http://mathoverflow.net/feeds/question/120463 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/120463/homogeneous-namba-like-forcing Homogeneous Namba-like forcing Trevor Wilson 2013-01-31T21:11:24Z 2013-02-01T00:47:41Z <p>Let $\kappa \ge \aleph_3$ be a regular cardinal that is countably closed ($\alpha^\omega &lt; \kappa$ for every $\alpha &lt; \kappa$.) I'm mostly interested in the case that $\kappa$ is strongly inaccessible. Can there be a homogeneous notion of forcing that makes $\text{cof}(\kappa^{+V}) &lt; \kappa$ without adding any bounded subsets of $\kappa$?</p> <p>If there is a Woodin cardinal above $\kappa$ then the stationary tower forcing could do this except that it is (probably) not homogeneous.</p> <p>If there is a forcing notion as desired then I believe the results of the paper "Stacking mice" would give a non-domestic mouse, so some large cardinals would be required to show that such a forcing exists. Can we get one from, <em>e.g.</em> a supercompact cardinal?</p>