出版者・発行元：京都大学数理解析研究所  

高さω_1 (最小非可算基数) の Souslin 木が Countable Support Iteration の極限
段階でどのような状況にあるか,その保存性に関する定理を発見し, また, この
応用について述べる。

出版者・発行元：京都大学数理解析研究所  

Morasses は単純な規則で記述され,複雑な様相を持つ。この
理論の整備, 新種 Morasses の考案, それらの相互関係に関する定理
を示す。

出版者・発行元：日本数学会  

半順序の族 Axiom C を立案し, 無限積を動的に行うようなも
のである Countable Support Iteration でこの族が閉じて
いることを示す。

出版者・発行元：神戸大学  

Countable Chain Condition (c.c.c.) を満たす半順序と
Countably Closed な半順序の Forcing 積を考え, 積の順序と
埋め込みについて定理を発見する。

出版者・発行元：Dartmouth College  

無限基数を対象とする組み合わせ論的な問題の解決のため,
Morasses や Iterated Forcing の手法を整備し, 別証明を
与える。

記述言語：英語   掲載種別：研究論文（大学，研究機関等紀要）  

出版者・発行元：南山大学  

Presented is a poset to force a simplified morass. The poset consists of finite symmetric systems. No fast functions are involved.

出版者・発行元：京都大学数理解析研究所  

Assuming three types of morasses, a map from the edges of the third uncountable cardinal into the set of all natural numbers is constructed. The map has no large homogeneous set.

出版者・発行元：南山大学  

We consider a poset to force a club subset of the second least
uncountable cardinal.
Two types of non-transitive elementry substructures of a transitive set
model of set theory are used.
One of them is of the size the least uncountable cardinal and the other
is countable.

出版者・発行元：南山大学  

出版者・発行元：南山大学  

This paper deals with a simple example of Aspero-Mota type iterated forcing of the length of the second uncountable cardinal.
The isomorphisms between models in the conditions are naturally lifted in the generic extensions. As a result, the Continuum Hypothesis (CH) is preserved
by every proper intermediate stage of the iteration. This generalizes a result of Aspero on CH and to be compared with that of Shelah.

出版者・発行元：Springer  

We present a new proper poset that forces the Mapping Reflection Principle of Moore.
The poset consists of finite conditions and enjoys amalgamations of several conditions at a time.

出版者・発行元：京都大学数理解析研究所  

We present a poset that is of finite in nature and yet adds a continous epsilon-chain of countable models of set theory. The length of the sequence is the least uncountable cardinal. As an application, we consider the Strong Reflection Principle of Todorcevic.

出版者・発行元：Polish Academy of Sciences  

This paper deals with a forcing axiom that is compatible with an entangled set. It turns out that the seemingly stronger notion of the entangledness here
is actually equivalent to the one found in the literatures. This is due to the referee. The construction is done by an Aspero-Mota type iterated forcing and it tends to be technical.

出版者・発行元：南山大学  

We deal with the negative Ramsey-type partition relation. We present a new proof of a result established by P. Erdos and R. Rado in 1956. We use a simplified morass formulated by D. Velleman in the 1980s.

出版者・発行元：南山大学  

This paper presents a simple example of iterated forcing that utilizes the newly developed Aspero-Mota method. Here we use only one transitive set universe and its countable elementary substructures.
For any elementary substructure with a finite set of markers, we interpret it as a substructure with a specified single initial segment of the sets of ordinals of the substructure.

出版者・発行元：南山大学経営学会  

We construct a function from the set of unordered pairs of the countable ordinals into a set that has exactly two elements 0 and 1.
The function is such that there exist neither uncountable 0-homogeneous sets nor 1-homogeneous sets of an order type omega + 2. We use a weakly nice simplified morass in the construction.

出版者・発行元：京都大学数理解析研究所  

We construct a notion of iterated forcing by side condition method. This notion of forcing closely follows the original countable support iteration of S. Shelah. We do not know whether the length of our iteration can be extended beyond the second uncountable cardinal.

出版者・発行元：Springer Berlin Heidelberg  

We show that a type of coding principle by J. Moore
is equiconsistent with a strongly inaccessible cardinal. We also show that a type of coding by S. Todorcevic entails the same large cardinal. We use simplified morasses by D. Velleman.

出版者・発行元：京都大学　数理解析研究所  

公理的集合論の推移的集合モデルの可算初等部分構造を取り上げる。それらを特徴づける各種の指標や同型、また、同型写像の拡張について紹介する。特に、集合論の各種仮説（連続体仮説、CHANGの予想など）との関係について簡潔な説明を与える。

出版者・発行元：京都大学　数理解析研究所  

We force a morass by a finite side condition method.

出版者・発行元：南山大学  

We plan to reproduce a type of Aspero-Mota iteration of finitely proper posets in two stages. In the first stage, we force a stationary set E of countable subsets
of the length of the iteration. In the second stage, we iteratively force with E-finitely proper forcing posets that satisfy a size restriction. In this note, we prepare fundamentals in this line of treatment of forcing. Subsequently we force the E. The remainder of our construction will be continued in a sequel to this note.

出版者・発行元：京都大学数理解析研究所  

We force a family of strongly almost disjoint functions by finite conditions in two stages. In the first stage, we forces a Kurepa tree by plain side conditions. In the second stage, we force the family by a ccc forcing poset that makes use of the Kurepa tree forced in the first stage. This explicates a role of side conditions in side condition methods.

出版者・発行元：南山大学  

We propose a second method to produce a morass from a matrix.　Our main idea is using a matrix instead of a generic filter.　We recursively attatch relevant parts of generic conditions to the　members of the matrix such that the whole structure that is formed　is a morass.

出版者・発行元：京都大学数理解析研究所  

We formulate a matrix with coherent sequences.　
A matrix comprises models of the set theory of a size equal to the least uncountable cardidinal.　A matrix with coherent sequences entails a simplified morass with linear limits.

出版者・発行元：南山大学  

We formulate matrices of isomorphic models of set theory with coherent sequences.
We show that these matrices entail simplified morasses with short linear limits.

出版者・発行元：京都大学数理解析研究所  

We force matrices of isomorphic models of set theory by countable conditions.
We show that the matrices entail semimorasses and morasses depending on the forcing method.
We also present a construction of a Souslin tree that makes use of a matrix.

出版者・発行元：南山大学  

Aspero and Mota find a new forcing axiom. Their axiom is more comprehesive than the Martin's Axiom. To establish the new axiom, they first force a matrix of
isomorphic models of set theory. Thereafter, they iterate proper forcing. We investigate the matrix in the first stage of the construction. We show that the matrix entails a Kurepa tree. We may further force a closed and unbounded subset of the least uncountable cardinal so that the strengthened matrix entails a quagmire. We also propose a matrix that entails a square principle.

出版者・発行元：京都大学数理解析研究所  

We show that the forcing axiom PFA(S) implies MRP. We also investigate weak club guessing in the forcing extensions via a type of Suslin trees.

出版者・発行元：京都大学数理解析研究所  

We formulate an iterated proper forcing with side conditions. We start with a measurable cardinal. Then we construct our iterated proper forcing along a squence of increasing transitive models of set theory. The measurable cardinal is turned into the second uncountable cardinal. This is a rendition of one of Neeman's original construction with two types of models.

出版者・発行元：南山大学  

We propose a couple of forcing axioms that have combined strengths of Martin's axiom and Chang's conjecture. We consider proper and semiproper forcings. Our new forcing axiom do not deal with generic conditions but semi-generic conditions, even when proper partial orders are at hand.

出版者・発行元：京都大学数理解析研究所  

We propose a forcing axiom that holds in the generic extensions by a Levy collapse. This collapse turns a measurable cardinal into the second uncountable cardinal. We show that Chang's conjecture and an instance of strong non-reflection hold by forcing over the ground model where this new forcing axiom is assumed to hold.

出版者・発行元：南山大学  

We construct set theory model by forcing not only where Chang's Conjecture holds but also where a non-reflecting stationary set exists. In particular, this
separates stronger types of Chang's Conjecture from the ordinary Chang's Conjecture. We make use of the skinniest stationary sets, a system of ladders and an associated system of closed unbounded sets.

出版者・発行元：京都大学数理解析研究所  

We formulate a class of preorders which we call preproper. This class contains the class of proper preorders and is a modification to the subproper preorders by R. Jensen. We show that this class of preorders is iterable under a type of revised countable support iteration.

出版者・発行元：南山大学  

We consider a notion of forcing which changes the cofinality of the second uncountable cardinal to omega.
This notion is a variant of the one due to R. Jensen.
We do not use infinitary languages to form ours.

出版者・発行元：京都大学数理解析研究所  

We show that the consistency strength of the Fodor-type Reflection Principle for the second uncountable cardinal is exactly that of a Mahlo cardinal.

出版者・発行元：南山大学  

We investigate the preservation of weak segments of the Proper Forcing Axiom (PFA) under omega_2-closed forcing. It turns out that we need a stronger segment
for a weaker one to hold in the generic extensions.

出版者・発行元：京都大学数理解析研究所  

We show there is no maximal forcing axiom compatible with tail club guessing.
On the other hand, we may formulate a maximal forcing axiom compatible with a type of weak club guessing

出版者・発行元：京都大学数理解析研究所  

We consider a principle which not only negates weak club guessing but also codes every subset of the least uncountable cardinal. In particular, the continuum hypothesis must fail under this principle

出版者・発行元：南山大学  

We deal with lots of elementary substructures of transitive sets which are models of fragments of set theory. We want to know more on iterated forcing which adds no new reals at limit stages.

出版者・発行元：南山大学  

We show a weak form of reflection principle combined with tail club guessing
negates a class of weak squares.

出版者・発行元：京都大学数理解析研究所  

We formulate a weak form of reflection. This principle is compatible with tail club guessing
on the first uncountable cardinal and implies the corresponding tail club guessing ideal is saturated.
We consider a forcing axiom which implies our principle and is compatible with tail club guessing via
a semiproper iterated forcing.

出版者・発行元：南山大学  

Assuming the diamond, we construct a notion of forcing which iterates Souslin trees. This iteration codes any family of iteratively generic cofinals paths by a single real.

出版者・発行元：京都大学数理解析研究所  

We formulate a principle, called ＼tau_{AC}, which implies both ＼psi_{AC}
and ＼phi_{AC}. we also force ＼tau_{AC} and conclude equiconsistencies of these.

出版者・発行元：Institute for Mathematical Scienses, National University of Singapore  

The Strong Reflection Principle (SRP) negates the continuum hypothesis (CH). To iteratively force SRP,
we may use notions of forcing which add no new countable sequences of ordinals. Hence the iteration
must add new reals at limit stages. We explicate this phenomenon with an example.

出版者・発行元：京都大学数理解析研究所  

We formulate combinatorial principles concerning the least and the second least uncountable cardinals. They fit between the Kurepa hypothesis and the weak diamonds due to H. Woodin. We separate some of the principles via forcing.

出版者・発行元：南山大学  

We investigate principles which fit between the Kurepa Hypothesis and the weak Kurepa Hypothesis

出版者・発行元：南山大学  

無限基数に関する組み合わせ論的命題phi_{AC}の無矛盾性の証明をiterated forcingにより示す。

出版者・発行元：南山大学  

S. Shelah による高次の無限基数に対応する Forcing Axiom が成立しないという定理があるが, ここでは, 二番目の非可算基数上においてより簡潔な証明を与える。

出版者・発行元：京都大学数理解析研究所  

Bounded Semiproper Forcing Axiom と Measurable Cardinal を前提とする
ことにより, S. Todorcevic の発見した組み合わせ論的原理 Theta_{AC}
の強化をはかる。

出版者・発行元：南山大学  

Templates に沿っての Iterated Forcing の手法は S. Shelah により開発された。
ここでは, この手法の基礎的な部分についてまとめる。

出版者・発行元：京都大学数理解析研究所  

^ωω-bounding であり Semiproper である Preorders の族が Simple Iteration
で閉じていることを示す。

出版者・発行元：南山大学  

最小非可算基数上の Nonstationary Ideal を Completely Bounded にする
Notion of Forcing の組み合わせ論的性質と巨大基数 につての分析を行う。
名古屋大学集合論セミナーでの結果をまとめる。

出版者・発行元：日本数学会トポロジー分科会General Topology  

Random Reals を付け加えない Semiproper Preorders を Simple Iterations
が保存することを示す。

出版者・発行元：南山大学  

Square Principle と呼ばれる複数の無限基数の関係した組み合わせ論的な命題の無矛盾性の証明に関係し, 定理を発見する。

出版者・発行元：南山大学  

名古屋大学集合論セミナーで得た Levy Collapses と Silver
Collapses, Generalized Prikry Forcing, および Laver
Forcing についての結果をまとめる。

出版者・発行元：南山大学  

M. Fitting の教科書に載っていた定理で, 命題論理におけ
る定理の自動証明の基礎となるものに, 集合論的な手法を
使った簡潔な別証明を与える。

出版者・発行元：南山大学  

Proper 半順序の族が Countable Support Iteration で保存されるという S. Shelah の理論の整備拡張を試みる。

出版者・発行元：神戸大学大学院  

ブール代数を使って, 公理的集合論 ZFaa が無矛盾ならば,
ZFaa に選択公理を付け加えた集合論も無矛盾であることを証明する。

We presented a new method to force a closed unbounded subset of uncountable structures.

We presented a class of posets that are iterated in an Aspero-Mota style.

We represent a consistency proof of a partition relation originally established by S. Todorcevic. We use a so-called side condition method newly developed by Aspero-Mota. We also report that the partition relation gets negated by a weakly nice simplified morass.

We provide a new proof of a negative partition relation with the triples orginally established by Erdos and Rado in 1956. We make use of a simplified morass that was found by Velleman in 1984. It is known that the morass exists in ZFC. Hence the negative partition relation holds in ZFC.

It is consistent that no Souslin trees but a non-special Aronszajn tree exists. We use a side condition method.

We report a forcing poset that forces what we call a morass-type matrix. A condition of the poset is represented by a pair of a finite symmetric system of Aspero-Mota and a finite function from the finite symmetric system into the least uncountable cardinal with some restrictions. We are forcing the rank function of the matrix and the matrix itself, simultaneously. We know that any morass-type matrix entails a simplified morass of D. Velleman.

初等部分構造を導入し、この構造を中心に、公理的集合論における構成の特色について話す。デルタシステムに関する簡単な構成を例示し、初等部分構造の縦方向と横方向へのそれぞれの拡張、および、それに伴う同形写像の持ち上げについて考える。特に、連続体仮説との関係について、整理し、最新の結果や応用について紹介する。

We formulate matrices with coherent sequences. We observe that these matrices entail morasses with linear limits. In particular, we have a construction of squares via these matrices. This direct construction is based on Velleman's original.

We claim that matrices of isomorphic models of set theory entail morass-like structures such as Kurepa trees, quagmires, square principle, semimorasses, and morasses. We also propose a type of completeness that leads to a simple construction of a Souslin tree via a matrix.

We consider a form of forcing axiom which has a sort of combined strength of Chang's Conjecture and an internal genericity. We study the effects of this forcing axiom with the Cohen forcing in this talk. We observe that the size of the set of reals and stronger forms of Chang's Conjecture are related under this forcing axiom.

It is shown by H. Sakai that Chang's Conjecture and a form of weak square hold　in the two stage generic extensions constructed by first forcing with the Levy collapse which turns a measurable　cardinal into the second uncountable cardinal and then forcing the weak square over the first extensions. We extract a form of forcing axiom　which captures an important property of the first extensions and show that Chang's Conjecture and an instance of strong non-reflection hold　by forcing over the ground model where the forcing axiom is assumed.

We show that the consistency strength of the Fodor-type Reflection Principle (FRP) for the second uncountable cardinal is exactly that of a Mahlo cardinal.

We show that a set theory with a type of coding by J. Moore is equiconsistent with the set theory with a strongly inaccessible cardinal.

We show there is no maximal forcing axiom compatible with tail　club guessing. On the other hand, we may formulate a maximal forcing axiom compatible with a type of weak club guessing.

We consider a principle which negates club guessing on the first uncountable cardinal. This principle decides the size of the power set of the set of natural numbers.

We review various types of stationary sets and related facts. We identify stationary sets involved in Part one. We give our iteration theorem in this area of set theory.

We consider SRP-type statements which follow from forcing axioms compatible with club guessing. The relevant p.o.sets are proper and some degree of semiproperness are satisfied to preserve a club guessing sequence.

We formulate a principle, called tau_{AC}, which implies both psi_{AC} and phi_{AC}. We also force tau_{AC} and conclude equiconsistencies of these.

We consider various types of weak Kurepa Hypotheses and diamond principles. We provide some consistency proofs via forcing.

We consider various properties of the projections of the ternary relation CODE. We observe they depend on the universes of set theory.

CH(連続体仮説), CB(complete bounding)およびsaturationの否定が成立する集合論のモデルをiterated forcingにより 構成する。

Bounded Semiproper Forcing Axiom と Measurable 基数の 存在を仮定するとき, S. Todorcevic の Theata_{AC} を 強くした無限基数に関する組み合わせ論的な原理が 導かれることを発表。

If we iteratively force withα-semiproper preorders by taking a suitable limit, then the iterated forcing enjoys the same property

We force with the fast function forcing at a localized reflecting cardinal. We show that the large cardinal has a Laver-type diamond sequence in the generic extensions.

We classify collections of countable subsets S of a fixed uncountable set. This is done in terms of the notion of forcing which shoots a club through S.