Sets of reals with cardinality between N0 and 2^N0? - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-06-19T04:51:19Z http://mathoverflow.net/feeds/question/48417 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/48417/sets-of-reals-with-cardinality-between-n0-and-2n0 Sets of reals with cardinality between N0 and 2^N0? anonymous 2010-12-06T03:36:48Z 2010-12-06T17:04:49Z <blockquote> <p><strong>Possible Duplicate:</strong><br> <a href="http://mathoverflow.net/questions/10227/in-set-theories-where-continuum-hypothesis-is-false-what-are-the-new-sets" rel="nofollow">In set theories where Continuum Hypothesis is false, what are the new sets?</a> </p> </blockquote> <p>If ZFC+not(CH) is consistent, there should be sets of real numbers with cardinality strictly between \$\aleph_0\$ and \$2^{\aleph_0}\$. Then why hasn't someone constructed a set of real numbers of intermediate cardinality in a model of ZFC+not(CH)?</p> <p>(I assume there are good reasons why this would be hard, so I'm asking what those reasons are rather than suggesting that I've come up with a new angle of attack...)</p>