ultrafilters' succession - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T19:43:38Z http://mathoverflow.net/feeds/question/84189 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/84189/ultrafilters-succession ultrafilters' succession unknown (google) 2011-12-23T20:26:25Z 2011-12-23T22:18:35Z <p>hi</p> <p>I'n looking for a increasing and bounded ultrafilters' succession in natural numbers with Rudin-Keisler order, actually I need to prove there is that succession the idea is </p> <p>$U_1,U_2,....$ with $U$ supreme and for all n $U_n &lt; U$ for $h_n$ in Rudin-Keisle order</p> <p>and por all n $U_n &lt; U_{n+1}$ for $g_n$ in Rudin-Keisle order</p> <p>and $h_n(m)= g_{n+1}/ocircle h_{n+1} (m)$ with m a natural number </p> <p>I have no idea how build the $h_n$ funtions</p> <p>thanks, sorry for my awful english </p> http://mathoverflow.net/questions/84189/ultrafilters-succession/84191#84191 Answer by Andreas Blass for ultrafilters' succession Andreas Blass 2011-12-23T22:14:00Z 2011-12-23T22:14:00Z <p>Most of your question sounds as if you just want an increasing $\omega$-sequence $(U_n)$ in the RK-order, with an upper bound $U$. Although you want these ultrafilters to be on $\omega$, it's a little easier to see the idea for a construction if you use some other countable sets as follows; you can always transfer the result to $\omega$ by suitable bijections. Let $S$ be the set of all those $\omega$-sequences $(a_k)$ of natural numbers in which all but finitely many components $a_n$ are zero. For each natural number $n$, let $h_n:S\to\omega^n$ be the "truncation" function that sends any $(a_k)\in S$ to the tuple of its first $n$ components <code>$(a_k)_{k&lt;n}$</code>. Notice that <code>$h_n=h_{n+1}\circ p_n$ where </code>$p_n:\omega^{n+1}\to\omega^n$ is the projection to the first $n$ components. Let $\mathcal X$ be the collection of those subsets $X\subseteq S$ such that, for some $n$, $p_n$ is one-to-one on <code>$h_{n+1}(X)$</code>. After checking that no finitely many sets in $\mathcal X$ cover all of $S$, we can let $U$ be any ultrafilter on $S$ that is disjoint from $\mathcal X$, and we can let <code>$U_n=h_n(U)$</code>. Then $p_n$ sends <code>$U_{n+1}$</code> to $U_n$ and is not one-to-one on any set in $U_n$ (because $U$ contains no set from $\mathcal X$). So $p_n$ witnesses that <code>$U_n&lt;_{RK}U_{n+1}$</code>. And of course the $h_n$'s witness that $U$ is an upper bound for all the $U_n$'s.</p> <p>I worry, though, that the word "supreme" in the question might mean that you want $U$ to be not only an upper bound but a <em>least</em> upper bound for the $U_n$'s. I don't know how to achieve that. </p> http://mathoverflow.net/questions/84189/ultrafilters-succession/84192#84192 Answer by Joel David Hamkins for ultrafilters' succession Joel David Hamkins 2011-12-23T22:18:35Z 2011-12-23T22:18:35Z <p>You don't say what the context is, but let's suppose at first that the question is asked in the large cardinal context of $\kappa$-complete ultrafilters on a measurable cardinal $\kappa$, which is a context where one often considers such questions. </p> <p>The first thing to say is that there may be no such example, even when there is a measurable cardinal. For example, in the canonical inner model $L[\mu]$ with one measurable cardinal $\kappa$, there is exactly one normal measure $\mu$ on $\kappa$, and every $\kappa$-complete ultrafilter on $\kappa$ is Rudin-Kiesler equivalent to a finite power $\mu^n$ of $\mu$. In particular, although these powers do indeed form an increasing chain $$\mu^1\lt_{RK}\ \mu^2\lt_{RK}\ \mu^3\lt_{RK}\ \cdots$$ in the Rudin-Kiesler order, there can be no measure on top of this chain as you request. In short, it is consistent with a measurable cardinal that the height of the Rudin-Kiesler order is $\omega$. </p> <p>From a larger large cardinal assumption, however, one can achieve higher Rudin-Kiseler ranks. For example, if $\kappa$ is $\kappa+2$-strong, then there must be ultrafilters $U$ on $\kappa$ with Rudin-Kiesler rank $\omega$, giving rise to the situation of your question. To see this, let $j:V\to M$ with $V_{\kappa+2}\subset M$ and we may assume $M^\kappa\subset M$. Let $X_1=\{j(f)(\kappa)\mid f:\kappa\to V\}$ be the seed hull of $\kappa$, which is elementary in $M$ and collapses to the ultrapower $j_1:V\to M_1$ of the induced normal measure $U_1$ generated by $j$. Consider the first ordinal missing from $X_1$, call it $\delta_1$, and let $X_2$ be the see hull $\{j(f)(\delta_0,\delta_1)\mid f:\kappa^2\to V\}$, using $\delta_0=\kappa$, and similarly define $\delta_n$ for each $n$. The ultrafilter $U_n$ induced by $A\in U_n\iff (\delta_0,\ldots,\delta_{n-1})\in j(A)$ is Rudin-Kiesler below $U$, and they form an increasing chain as desired. The chain is strictly increaing precisely because no $X_n$ can exhaust all of $M$ below $j(\kappa)$. </p> <p>In the general case, or even in the case of ultrafilters on $\omega$, it remains easy to build increasing chains in the Rudin-Kiesler order. For example, the successive products of a fixed ultrafilter form a strictly increasing chain $$U\lt_{RK} U^2\lt_{RK} U^3\lt_{Rk}\cdots$$ To place an ultrafilter $U$ on top, however, takes some more delicate work (and I see now that Andreas Blass has posted an answer indicated how to undertake that delicate work). </p>