Logical relationships between weakenings of AC - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T16:30:00Z http://mathoverflow.net/feeds/question/96108 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/96108/logical-relationships-between-weakenings-of-ac Logical relationships between weakenings of AC Everett Piper 2012-05-06T03:38:42Z 2012-05-06T03:38:42Z <p>What are the known logical implications between weak choice principles like <code>$DC_\kappa$", the</code>ultrafilter theorem for sets of size $\kappa$" (by which I mean every filter over a set A of size $\kappa$ can be extended to a non-principle ultrafilter), and every relation $R$ over $A$ can be uniformized by a function $f$ with the same domain as $R$"? </p> <p>Jech's book on the Axiom of Choice gives nice proofs that uniformization for $A$" and "the ultrafilter theorem for sets of size $|A|$ logically independent, i.e. neither implies the other. My concern is how either of these consequences of AC relate to uniformization. </p> <p>I assume $DC_\kappa$ does not imply uniformization of sets of size larger than $\kappa$ but I have yet to see any proof of this fact. </p> <p>Are the three principles logically independent of one another on a global scale? How do they relate on a more local level? </p>