SET-THEORY MEETING in KASUGAI

July 21–22, 2001

中部大学一号館第二小会議室

PROGRAM

\underline{7 月 21 日（土）} 14:00 – 15:00 矢田部俊介（神戸大学） Forcing indestructibility of MAD families

15:30 – 16:30

{高橋真（神戸大学） \atop + 吉信康夫（名古屋大学）}

Banach-Mazur games on Boolean Algebras and σ-short Boolean algebras

17:00 – 18:00 村上雅彦（法政大学） Canonical generic extension of a small nonstandard universe

l9:00 –    会食  (ステーキのあさくま – 添付地図参照)

\underline{7 月 22 日（日）} 09:30 – 10:30 依岡輝幸（神戸大学） The cofinality of the strong measure zero ideal

10:45 – 11:30 江田勝哉（早稲田大学）非可換超積

昼食

13:00 – 13:45 渕野昌（中部大学）関数の連続性について
14:00 – 14:45 塩谷真弘（筑波大学）  Gitik’s theorem revisited

15:15 – 16:l5 宮元忠敏（南山大学） On the complete boundingness and the strong reflection principle

ABSTRACTS

Banach-Mazur games on Boolean Algebras and σ-short Boolean algebras

Makoto TAKAHASHI 神戸大学 発達科学部 makoto@kobe-u.ac.jp  and Yasuo YOSHINOBU 名古屋大学大学院 人間情報学研究科 yosinobu@math.nagoya-u.ac.jp

In this talk we consider the Banach-Mazur game G(B) on an atomless Boolean algebra B(or separative poset P) in which two players Empty and Nonempty choose alternately nonzero elements of B so that the resulting play is a descending sequence of elements. Nonempty wins if the sequence has a nonzero lower bound. We say that a Boolean algebra B is game-closed if Nonempty has a winning strategy in G(B). Jech-Shelah showed that it is consistent to ZFC that there exists a game-closed Boolean algebra which does not have a σ-closed dense subset. What can we say about Empty’s winning strategies. We say that a dense subset D of a Boolean algebra B is σ-short if every strictly decreasing sequence of length ω in D does not have a nonzero lower bound in B. A Boolean algebra B is said to be σ-short if it has a σ-short dense subset. It is easy to see that if the Boolean completion of B is σ-short, then Empty has a winning strategy in G(B). We shall consider whether the converse is true or not. Free Boolean algebras and measure algebras are examples of σ-short Boolean algebras. We shall show that regularly filtered Boolean algebras are σ-short but the converse is not true even if they are ccc.

Cofinality of the ideal SN

Teruyuki Yorioka Graduate School of Science and Technology Kobe University Rokkodai, Nada-ku, Kobe 657-8501, Japan yorioka@kurt.cla.kobe-u.ac.jp

We say that a subset X of the set of reals ℝ has strong measure zero if for any countable sequence ⟨ε_{n}; n ∈ ω⟩ of positive real numbers, there is a countable sequence ⟨I_{n}; n ∈ ω⟩ of basic open sets such that X ⊆ ⋃_{n∈ω} I_{n} and μ(I_{n}) ≤ ε_{n} for all n ∈ ω.

For an ideal I on ℝ, let the cofinality cof(I) of I be the smallest power of a base of I. And for regular cardinal κ, let 𝔟_{κ} and 𝔡_{κ} be the unbounding and dominating number for κ^{κ} respectively. Clearly κ < 𝔟_{κ} ≤ 𝔡_{κ} ≤ 2^{κ} holds.

For functions f and g in ω^{ω}, define f ≫ g if it’s true that ∀k ∈ ω∃M ∈ ω∀n ≥ M(f(n) ≥ g(n^{k})). We say that a subset A of 2^{ω} is f-small if there is a τ ∈ (2^{<ω})^{ω} such that, letting g_{τ} (n) = |τ(n)| for all n ∈ ω, g_{τ} ≫ f and A ⊆ ⋂_{m∈ω} ⋃_{n≥m}[τ(n)]. Then for any dominating family F, a subset A has strong measure zero iff A is f-small for all f ∈ F.

Let SN be the set of all strong measure zero subsets of ℝ. Clearly, SN is a σ-ideal but it is not necessarily a Borel ideal, hence the cofinality of SN may be larger than the power 𝔠 of the continuum.

We prove that, assuming CH, the cofinality of the ideal SN is equal to 𝔡_{ℵ_{1}}. Therefore the cofinality of SN is larger than 𝔠 under CH.

More generally, if 𝔡 = ℵ_{1} and cof(f-small) = ℵ_{1} for all f in ω^{ω}, then cof(SN) = 𝔡_{ℵ_{1}} holds.

It follows that it’s consistent with ZFC that the cofinality of SN is less than 𝔠.

Even if both 𝔠 and 𝔡 are larger than ℵ_{1}, we have similar statements.

If cov(M) = 𝔡 and there exists a dominating family {f_{α}; α < 𝔡} in ω^{ω} such that add(f_{α}-small) = cof(f_{α}-small) = 𝔡 for all α < 𝔡, then cof(SN) = 𝔡_{𝔡} holds.

関数の連続性について

渕野昌 (Sakaé FUCHINO) 中部大学理学教室 fuchino@isc.chubu.ac.jp

Blumberg の定理 (1922) により，任意の関数 f : ℝ → ℝ に対し， dense な D ⊆ ℝ で f |` D が連続になるようなものが存在する．

ここでの “dense な” を “uncountable な”, “perfect set を含む”, “non meager な”, “non measure 0” などで置き換えて得られる命題は集合論から独立であることが知られている — “perfect set を含む” で置き換えて得られる命題は ZFC とは inconsistent だが Solovay model (of ZF + DC + every set of real is measurable) で成り立ち， ZF + DC + AD での定理となる．

本講演では，これらの命題に関連したいくつかの結果を概観し， Fuchino & Shelah による関連の結果にも触れる．

Gitik’s theorem revisited

塩谷真弘 (Masahiro SHIOYA) 筑波大学数学系 shioya@math.tsukuba.ac.jp

I will outline the proof of the following

Theorem. It is consistent relative to κ κ^{+}-supercompact, that some stationary subset of P_{κ}κ^{+} cannot split into κ^{+} stationary sets.

On The Complete Boundingness and The Strong Reflection Principle

宮元忠敏 (Tadatoshi MIYAMOTO) 南山大学数理情報学部数理科学科 miyamoto@nanzan-u.ac.jp

Let S be a set of countable subsets of a given set A which contains the first uncountable cardinal. We consider a notion of forcing P which shoots a club through this S. It turns out that there is a simple logical connectaion between the richness of S and the niceness of P. For example, S is projectively stationary iff P preserves the stationary subsets of the first uncountable cardinal.

We move on to consider the Strong Reflection principle (SRP) and weak forcing axioms which correspond to the consequences of SRP. A very weak forcing axiom for some of the P as above together with a form of weak Chang’s Conjecture is equivalent to the Complete boundingness (CB).