An eventually different function adding no Solovay real nor dominating function? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T15:46:12Z http://mathoverflow.net/feeds/question/41447 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/41447/an-eventually-different-function-adding-no-solovay-real-nor-dominating-function An eventually different function adding no Solovay real nor dominating function? Bjørn Kjos-Hanssen 2010-10-07T19:42:11Z 2010-10-14T08:35:17Z <h1>Definitions</h1> <p>I believe set theorists have studied all of the following three notions in the context of forcing extensions of a model of ZFC, $M$ (hopefully the terminology is the standard one).</p> <ol> <li><p>A function $f:\mathbb N\rightarrow\mathbb N$ is <em>eventually different</em> if for each function $g:\mathbb N\rightarrow\mathbb N$, $g\in M$, the set <code>$\{n: f(n)=g(n)\}$</code> is finite.</p></li> <li><p>A real $r\in [0,1]$ is a <em>Solovay random real</em> if for each measure-zero subset $S$ of $\mathbb R$ with $S\in M$, we have $r\not\in S$.</p></li> <li><p>A function $f:\mathbb N\rightarrow\mathbb N$ is <em>dominating</em> if for each function $g:\mathbb N\rightarrow\mathbb N$ in the ground model $M$, the set <code>$\{n: f(n)\le g(n)\}$</code> is finite. <br></p> <h1>Motivation</h1></li> </ol> <p><br> An eventually different function that is not too fast-growing is reminiscent of a random real. Can we always use it to construct a random real? The analogous problem in computability theory was quite difficult but has been solved by Kumabe and Lewis (J. LMS, 2009). </p> <h1>Question</h1> <blockquote> <p><strong>I.</strong> Is it possible to add an eventually different function to $M$ while adding neither a Solovay real nor a dominating function? <br></p> </blockquote> <p>EDIT: Now stating the question in the strongest possible form, which is the one Andrés Caicedo answers below.</p> http://mathoverflow.net/questions/41447/an-eventually-different-function-adding-no-solovay-real-nor-dominating-function/41461#41461 Answer by Andres Caicedo for An eventually different function adding no Solovay real nor dominating function? Andres Caicedo 2010-10-07T21:54:51Z 2010-10-07T21:54:51Z <p>Hi Bjørn, and congratulations to you and Bonnie!</p> <p>The answer to I is yes. In fact, there is a standard way of doing this, with the "eventually different forcing ${\mathbb E}$". This notion does not add random or dominating reals, and adds an eventually different function. </p> <p>Conditions have the form $(s, A)$ where $s\in\omega^{&lt;\omega}$ and $A\in[\omega^\omega]^{&lt;\omega}$, with $(s, A)\le(s',A')$ iff $s\supseteq s'$, $A\supseteq A'$, and for all $f\in A'$ and $j\in[|s'|,|s|)$, we have $s(j)\ne f(j)$. (For me, $p\le q$ means that $p$ is stronger.) </p> <p>This is a nice forcing: It is ccc, in fact, $\sigma$-centered, since any two conditions with the same first coordinate are compatible. But no $\sigma$-centered forcing adds random reals. </p> <p>That ${\mathbb E}$ does not add dominating reals is a tad more work. But you can find a written proof in section 7.4.B of "Set Theory: On the structure of the real line", by Tomek Bartoszy´nski and Haim Judah. Let me know if you do not have access to a copy. </p>