Consistency Results Separating Three Cardinal Characteristics Simultaneously - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T23:32:58Z http://mathoverflow.net/feeds/question/37188 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/37188/consistency-results-separating-three-cardinal-characteristics-simultaneously Consistency Results Separating Three Cardinal Characteristics Simultaneously Justin Palumbo 2010-08-30T19:31:18Z 2010-08-31T10:06:02Z <p>(For information on cardinal characteristics of the continuum aka cardinal invariants see Joel David Hamkins' MO answer <a href="http://mathoverflow.net/questions/8972#9027" rel="nofollow">here</a>; Andreas Blass's <a href="http://www.math.lsa.umich.edu/~ablass/hbk.pdf" rel="nofollow">handbook article</a> is an excellent reference.)</p> <p>Problem 2.3 of Shelah's <a href="http://arxiv.org/abs/math/9906113" rel="nofollow">"On What I Do Not Understand (and Have Something to Say), Part I"</a> (published in 2000 in Fundamenta Mathematicae) states, "Investigate cardinal invariants of the continuum showing $\geq 3$ may have prescribed order". One major barrier to such an investigation is the fact that countable support iteration of proper forcings yields models where the continuum is $\aleph_2$. In such models given any three cardinal characteristics at least two will have to be equal.</p> <p>My question is the following. To what extent has such an investigation been pursued? In either the literature or folklore are there any results proving the consistency of inequalities $\mathfrak{c}_0&lt;\mathfrak{c}_1&lt;\mathfrak{c}_2$ where the $\mathfrak{c}_i$ are cardinal characteristics?</p> http://mathoverflow.net/questions/37188/consistency-results-separating-three-cardinal-characteristics-simultaneously/37233#37233 Answer by Stefan Geschke for Consistency Results Separating Three Cardinal Characteristics Simultaneously Stefan Geschke 2010-08-31T08:52:44Z 2010-08-31T10:06:02Z <p>There is a <a href="http://front.math.ucdavis.edu/math.LO/9205208" rel="nofollow">paper</a> of Shelah and Goldstern devoted to the separation of many simple cardinal invariants (this is a technical term). There are more recent papers on this subject by Kellner and Shelah, if I remember correctly. </p> <p>An easy case that I am very familiar with are the so called localization numbers. A closed set $S\subseteq\omega^\omega$ is $n$-ary if in the tree $T(S)$ of finite initial segments of elements of $S$ every node has at most $n$ immediate successors.<br> For $n\geq 2$ let $\ell_n$ be the least size of a family of $(n-1)$-ary sets that covers all of $n^\omega$.</p> <p>Any finitely many $\ell_n$ can be separated from each other simultaneously.<br> This is shown in [Geschke, Kojman, Convexity numbers of closed subsets in R^n, Proc. Am. Math. Soc. 130, No. 10, 2871-2881 (2002)], which is <a href="http://www.hausdorff-center.uni-bonn.de/people/geschke/publications" rel="nofollow">here</a>.</p> <p>Proofs of such statements usually involve forcing with large countable support products over a model of GCH rather than iterated forcing. However, there are also some examples that use iterated forcing. See for example <a href="http://arxiv.org/abs/math/9712288" rel="nofollow">this paper</a> by Shelah and Steprans.</p>