Antichains of Cardinals in ZF Without Choice... - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T18:05:24Z http://mathoverflow.net/feeds/question/82972 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/82972/antichains-of-cardinals-in-zf-without-choice Antichains of Cardinals in ZF Without Choice... Asher M. Kach 2011-12-08T15:09:36Z 2011-12-09T06:35:46Z <p>With the Axiom of Choice, the cardinals form a nice linearly ordered "set". In the absence of the Axiom of Choice, the cardinals form a partially ordered "set". Broadly, I am wondering what properties these posets can have.</p> <p>A specific question I am interested in is the following.</p> <p><b> Is there, for each (infinite) subset $S \subseteq \mathbb{N}$ containing $1$ and not containing $0$, a model of ZF in which there is a maximal antichain of cardinals of size $n$ if and only if $n \in S$? </b></p> <p>Though I have a mild interest in knowing how such a model would be constructed (assuming a positive answer), my primary interest is in knowing that it is (is not) possible. Hence, if such a result exists in the literature, a citation would be all that I ask for.</p>