most recent 30 from http://mathoverflow.net 2013-05-20T14:46:29Z http://mathoverflow.net/feeds/question/64280 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/64280/a-question-about-aleph-1-closed-forcing-notions Mohammad 2011-05-08T10:35:27Z 2011-09-04T05:48:12Z

<p>Is it consistent that every non-trivial $\aleph_1-$closed forcing notion of size $2^{\aleph_0}$ (which is trivially the minimal size of such forcing notions) collapses some cardinals? (we can ask the same question for larger cardinals.)</p> <p>Remark: This statement follows trivially from "Foreman`s maximality principle" which states that every non-trivial forcing notion either adds a real or collapses some cardinals. As far as I know the consistency of this principle is unknown.</p>

http://mathoverflow.net/questions/64280/a-question-about-aleph-1-closed-forcing-notions/73254#73254 Norman Lewis Perlmutter 2011-08-20T00:40:49Z 2011-08-20T00:40:49Z

<p>I can't answer your question, but I can give a simpler sounding formulation that might be helpful. Analyze the question in two cases.</p> <p>Case 1: The continuum hypothesis holds <br> In this case, the statement is false, because any $&lt;\aleph_1$ -closed forcing of size $\aleph_1$ cannot collapse cardinals. The forcing to add a cohen subset to $\aleph_1$ is a nontrivial example of such a forcing. </p> <p>Case 2: The continuum hypothesis fails <br> In this case, it is a theorem that every $&lt;\aleph_1$-closed forcing notion which collapses a cardinal collapses the continuum. (See <a href="http://mathoverflow.net/questions/11633/is-it-possible-for-countably-closed-forcing-to-collapse-aleph-2-to-aleph-1%20%22this%20question%22" rel="nofollow">this question</a>, referenced by Joel in the comments.) But every such forcing is equivalent to the canonical collapse forcing to collapse the continuum to $\aleph_1$. The most general version of this latter theorem that I know of (although the degree of generality makes it hard to follow) can be found in Handbook of Boolean Algebras, Volume 2, Corollary 1.15. </p> <p>So really, your question boils down to whether every $&lt;\aleph_1$-closed forcing of size continuum is isomorphic to Coll$(\aleph_1, c)$ This sounds strange to me, but I can't prove it's false, and if Foreman is entertaining it, who am I to judge it?</p>