[logic-ml] 第5回松山TGSAセミナーのお知らせ

[Hiroshi Fujita]

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。

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第5回 松山TGSAセミナー
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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
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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。

-- 
藤田 博司 <fujita.hiroshi.mh at ehime-u.ac.jp>
愛媛大学理学部／大学院理工学研究科数理物質科学専攻