Vitali Sets vs Bernstein Sets... - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T18:15:54Z http://mathoverflow.net/feeds/question/71575 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/71575/vitali-sets-vs-bernstein-sets Vitali Sets vs Bernstein Sets... George Lazou 2011-07-29T13:51:23Z 2012-05-21T00:41:47Z <p>AC is enough to guarantee the existence of both Bernstein Sets and Vitali Sets...</p> <p>However is the existence of Vitali Sets strictly weaker than that of Bernstein Sets?</p> <p>What about the other way round?</p> http://mathoverflow.net/questions/71575/vitali-sets-vs-bernstein-sets/97453#97453 Answer by Liang Yu for Vitali Sets vs Bernstein Sets... Liang Yu 2012-05-20T02:14:28Z 2012-05-21T00:41:47Z <p>For your second definition of Vitali set, I have a weak partial answer. Namely the existence of a Bernstein set does not imply the existence of a \$T\$-Vitali set. The answer can be found in logic blog maintained by Andre Nies:<br> <a href="http://dl.dropbox.com/u/370127/Blog/Blog2012.pdf" rel="nofollow">http://dl.dropbox.com/u/370127/Blog/Blog2012.pdf</a></p> <p>Note that a Turing degree does not need to be an addition group.</p> <p>I don't know whether the existence of a Vitali set implies the existence of a Bernstein set. But it is not difficult to see, under \$ZF+DC\$, that there is a Vitali set (if it exists) which contains a perfect subset.</p> <p>For you first definition of Vitali set, I have no idea.</p>