Can $\omega_1$ be supercompact? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T09:12:37Z http://mathoverflow.net/feeds/question/103895 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/103895/can-omega-1-be-supercompact Can $\omega_1$ be supercompact? Trevor Wilson 2012-08-03T20:27:15Z 2012-08-03T20:35:00Z <p>Is "ZF + $\omega_1$ is supercompact" consistent relative to "ZFC + there is a supercompact cardinal"?</p> <p>In particular, if $\delta$ is supercompact, does it remain so in $V(\mathbb{R} \cap V[G])$ where $G \subset Col(\omega,&lt;\delta)$ is $V$-generic? This seems to be the case for measurability but I am having trouble proving it for supercompactness. It seems likely that someone else has tried this, so I though I'd ask here.</p> <p>The appropriate definition of supercompactness in ZF is the one in terms of normal fine measures, where normality is defined using diagonal intersections.</p> <p>I am aware that $\omega_1$ has some amount of supercompactness under AD. I am interested in a more direct proof using forcing, which I hope will give (full) supercompactness.</p> http://mathoverflow.net/questions/103895/can-omega-1-be-supercompact/103897#103897 Answer by Asaf Karagila for Can $\omega_1$ be supercompact? Asaf Karagila 2012-08-03T20:35:00Z 2012-08-03T20:35:00Z <p>While might not be a full and satisfactory answer, you might be interested in the following paper by Spector:</p> <blockquote> <p>Spector, M. <strong><a href="http://www.ams.org/mathscinet-getitem?mr=1134195" rel="nofollow">Iterated extended ultrapowers and supercompactness without choice.</a></strong>, <em>Ann. Pure Appl. Logic</em> <strong>54</strong> (1991), no. 2, 179–194. </p> </blockquote>