GCH+ Kurepa Families - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T15:15:04Z http://mathoverflow.net/feeds/question/84123 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/84123/gch-kurepa-families GCH+ Kurepa Families Ioannis Souldatos 2011-12-22T21:52:38Z 2012-08-06T06:21:06Z <p>I have a couple of questions about known theorems for GCH+Kurepa families.</p> <p>Definition first: Let $\kappa$ be a infinite cardinal. A $\kappa^+$ Kurepa family is a family $F$ of subsets of $\kappa^+$ such that $F$ has size $>\kappa^+$ and for every $a&lt;\kappa^+$, the set $\lbrace X\cap \alpha|X\in F\rbrace$ has size $\le\kappa$. </p> <p>(The definition can be given in terms of tress too).</p> <p>$KH(\kappa^+)$ is the statement that a $\kappa^+$ Kurepa family exists.</p> <p>Please correct me, if I am mistaken, but we know that $KH(\kappa^+)$ holds for all infinite $\kappa$ in $L$ (the constructible universe). Also, if $\lambda$ is an inaccessible cardinal and we collapse $\lambda$ to $\aleph_2$, then in the generic extension $KH(\aleph_1)$ fails. (Look also this <a href="http://mathoverflow.net/questions/68533" rel="nofollow">http://mathoverflow.net/questions/68533</a>)</p> <p>So, my questions are:</p> <p>1) Do we know any models where GCH holds and $KH(\kappa^+)$ fails for all $\kappa$? </p> <p>2) If this is not the case, can we at least have GCH+ the failure of $KH(\aleph_{\alpha+1})$, for <strong>all</strong> $\alpha$ <strong>countable</strong> ordinals?</p> <p>3) If (2) is not known either, then fix some $\alpha$ countable ordinal $>0$. Can we have GCH+ the failure of $KH(\aleph_{\alpha+1})$? </p> <p>4) If the ground model satisfies GCH, after we collapse an inaccessible cardinal to $\aleph_2$ do we still get GCH?</p> <p>I am sure if I am asking too much. I just want to see what we already know.</p> <p>PS. What is the right way to pronounce Kurepa? Is it KUrepa (stress on KU), or KuREpa (stress on RE), or KurePA?</p> http://mathoverflow.net/questions/84123/gch-kurepa-families/103804#103804 Answer by Philipp Schlicht for GCH+ Kurepa Families Philipp Schlicht 2012-08-02T18:06:58Z 2012-08-06T06:21:06Z <p>Hi Ioannis! I guess you might know the answer by now; if we suppose for 1 that there is a class of inaccessible cardinals in the ground model and force with an Easton support product of Lévy collapses between the inaccessibles, we obtain a model of GCH where $KH(\kappa^+)$ fails for all $\kappa$; the argument is the same as for $\omega_1$. Also the failure of $KH(\kappa^+)$ implies that $\kappa^{++}$ is inaccessible in $L$, so we need the inaccessibles. </p>