LOGICメーリングリストの皆さま、
愛媛大学の藤田です。
このほど下記のセミナーを愛媛大学理学部において開催することとなりましたの でお知らせいたします。
============================================== 第5回 松山TGSAセミナー ============================================== 日時: 2014年9月8日(月) 16時30分〜17時30分 会場: 愛媛大学理学部 2号館201号室 (数学科大演習室) 講演者: 南 裕明さん （神戸大学） 演題: Mathias-Prikry forcing and dominating reals ============================================== Abstract: For a countable set X, we call $\mathcal{I}$ an ideal on X if $\mathcal{I}$ is a family of subsets of X closed under the taking subsets and unions. We assume all ideals on X contains the family of finite subsets of X. The Mathias-Prikry forcing associated with an ideal $\mathcal{I}$ on a countable subset X, denoted by $\mathbb{M}_{\mathcal{I}}$, consist of pairs (s,A) such that s is a finite subsets of X, A in $\mathcal{I}$ and $s\cap A=\emptyset$. The ordering is given by $(s,A)\leq (t,B)$ if t is a subset of s and B is a subset of A and $(s\setminus t)\cap B=\emptyset$.
The Mathias-Prikry forcing adds a new subset of X which diagonalize ideal $\mathcal{I}$, that is, $\mathbb{M}_{\mathcal{I}}$ adds a new subset $\dot{A}$ of X such that $X\cap I$ is finite for every I in $\mathcal{I}$. So Mathias-Prikry forcing plays significant role when we investigate ultrafilter, ideal or mad family.
Some additional nice properties of the Mathias-Prikry forcing depends on $\mathcal{I}$. For example, $\mathcal{U}$ is a Ramsey ultrafilter if and only if $\mathbb{M}_{\mathcal{U}^{*}}$ does not add Cohen real.
The speaker and Michael Hru\v{s}'{a}k give a characterization of ideals $\mathcal{I}$ such that $\mathbb{M}_{\mathcal{I}}$ adds no dominating real. We say a forcing notion $\mathbb{P}$ adds dominating reals if $\mathbb{P}$ adds a new function $\dot{g}$ from $\omega$ to $\omega$ such that for $f\in\omega^{\omega}\cap V$, $f(n)<\dot{g}(n)$ for all but finitely many $n\in\omega$.
We show that $\mathbb{M}_{\mathcal{I}^{*}}$ adds dominating reals if and only if $\mathcal{I}^{<\omega}$ is $P^{+}$-ideal.
Recently, David Chodounsk'{y}, and Du\v{s}an Repov\v{s} and Lyubomyr Zdomskyy give another characterization of ideal $\mathcal{I}$ with covering property such that $\mathbb{M}_{\mathcal{I}}$ adds no dominating real. We will talk about recent development of these result and application. ==============================================
松山TGSAセミナー（Matsuyama Seminar on Topology, Geometry, Set Theory and their Applications)は、位相空間論・幾何学・集合論およびその関連分野 を広く扱うセミナーで、愛媛大学のスタッフが中心となって定期的に開催して います。今後の予定についてはWebサイト http://www.math.sci.ehime-u.ac.jp/MTGSA.html をご覧ください。