tag:blogger.com,1999:blog-11569812.post5665177943246316335..comments2023-05-24T06:22:25.402-07:00Comments on The Orange Juice Files: A note on ultrafilters and voting.Theohttp://www.blogger.com/profile/03344294173628793721noreply@blogger.comBlogger2125tag:blogger.com,1999:blog-11569812.post-57586557843554157002007-06-27T05:18:00.000-07:002007-06-27T05:18:00.000-07:00Re problem 2: Any countable nonprincipal ultraprod...Re problem 2: Any countable nonprincipal ultraproduct is aleph_1-saturated. Under CH, all countable nonprincipal ultrapowers of the reals have cardinality aleph_1 and are elementarily equivalent, and hence are isomorphic by saturation. I do think I've heard of the converse before, but I don't know where and certainly have no idea of the proof.Anonymoushttps://www.blogger.com/profile/17149368177108023725noreply@blogger.comtag:blogger.com,1999:blog-11569812.post-8834201581874809692007-06-26T12:05:00.000-07:002007-06-26T12:05:00.000-07:00In any case, the answer to Problem (1), that an ul...In any case, the answer to Problem (1), that an ultrafilter limit is not necessarily the limit of a large subsequence, is easy enough, when I remember how to do it.<BR/><BR/>Let me consider the following filter on ${\mathbb N}^2 = {\mathbb N}_1 \cup {\mathbb N}_2 \cup {\mathbb N}_3 \cup \dots$, where ${\mathbb N}_i$ is a copy of the natural numbers. I'll say that a set $A$ is large if, for Theohttps://www.blogger.com/profile/03344294173628793721noreply@blogger.com