Cofinality of normal ideals on P_κ(λ), I

理想的相对共尾性

Relative cofinality of ideals

该研究引入理想共尾性不变量的双参数修正版本，纳入理想对相互作用，确定ω上关键理想对的不变量值，二分实数线理想对并研究极大理想的相对共尾性，推进基类结构研究。

Canonical models for aleph_1 combinatorics

We define the property of Pi_2-compactness of a statement phi of set theory, meaning roughly that the hard core of the impact of phi on combinatorics of aleph_1 can be isolated in a canonical model for the statement phi. We show that the following statements are Pi_2-compact: ``dominating number = aleph_1,'' ``cofinality of the meager ideal = aleph_1'', ``cofinality of the null ideal = aleph_1'', existence of various types of Souslin trees and variations on uniformity of measure and category = aleph_1. Several important new metamathematical patterns among classical statements of set theory are…

关于强测度零集理想的共尾度与覆盖的进一步研究

More about the cofinality and the covering of the ideal of strong measure zero sets

本文通过新组合原理 $\mathrm{DS}(δ)$、关系系统乘积及力迫迭代，改进强测度零集理想 $\mathcal{SN}$ 的共尾度与覆盖数界，并证明多种基数特征分离的一致性。

I[omega_2] can be the nonstationary ideal on Cof(omega_1)

We answer a question of Shelah by showing that it is consistent that every set of ordinals of cofinality omega_1 in I[omega_2] is nonstationary if and only if it is consistent that that there is a kappa^+ Mahlo cardinal kappa.

Ideals, Cohen sets and consistent extensions of the Erdős-Dushnik-Miller Theorem

We present two different types of models where, for certain singular cardinals lambda of uncountable cofinality, lambda -> (lambda, omega+1)^2, although lambda is not a strong limit cardinal. We announce, here, and will present in a subsequent paper, that, for example, consistently, aleph_{omega_1} not-> (aleph_{omega_1}, omega+1)^2 and consistently, 2^{aleph_0} not-> (2^{aleph_0},omega +1)^2 .

非正规 Magidor-Radin 型力迫

Non-Normal Magidor-Radin Types of Forcings

本文发展 Magidor 和 Radin 力迫的非正规变体，证明其可添加 Cohen 泛型函数，但仅在非改变共尾性情形；并证明扩展器基力迫不添加新子集，且 Gitik 力迫变体可改变可测基数共尾性至 ω₁ 并引入 Cohen 子集。

The structure of pleasant ideals

Normal ideals on regular uncountable cardinals are familiar objects. We investigate ideals that are pleasant--while a normal ideal is closed under arbitrary diagonal unions, a pleasant ideal is closed only under diagonal unions indexed by sets that are elements of the ideal. We show any selective ideal extending the nonstationary ideal must be normal.

Large normal ideals concentrating on a fixed small cardinality

A property of a filter, a kind of large cardinal property, suffices for the proof in Liu Shelah [LiSh:484] and is proved consistent as required there. A natural property which looks better, not only is not obtained here, but is shown to be false. On earlier related theorems see Gitik Shelah [GiSh310].

简单、次直接不可约弱二补格

Simple, subdirectly irreducible weakly dicomplemented lattices

本文研究了弱二补格中正常滤子的性质，证明其格结构并特征化了简单、次直接不可约和正则WDLs。

PCF and infinite free subsets

We give another proof that for every lambda >= beth_omega for every large enough regular kappa < beth_omega we have lambda^{[kappa]}= lambda, dealing with sufficient conditions for replacing beth_omega by aleph_omega. In section 2 we show that large pcf(a) implies existence of free sets. An example is that if pp(aleph_omega)> aleph_{omega_1} then for every algebra M of cardinality aleph_omega with countably many functions, for some a_n in M (for n< omega) we have a_n notin cl_M({a_l: l not= n, l<omega}). Then we present results complementary to those of section 2 (but not close enough): if IND…

Dense ideals and cardinal arithmetic

From large cardinals we show the consistency of normal, fine, $κ$-complete $λ$-dense ideals on $\mathcal{P}_κ(λ)$ for successor $κ$. We explore the interplay between dense ideals, cardinal arithmetic, and squares, answering some open questions of Foreman.

On the universality of the nonstationary ideal

Burke \cite{MR1472122} proved that the generalized nonstationary ideal, denoted NS, is universal in the following sense: every normal ideal, and every tower of normal ideals of inaccessible height, is a canonical Rudin-Keisler projection of the restriction of $\text{NS}$ to some stationary set. We investigate how far Burke's theorem can be pushed, by analyzing the universality properties of NS with respect to the wider class of \emph{$\mathcal{C}$-systems of filters} introduced by Audrito-Steila \cite{AudritoSteila}. First we answer a question of \cite{AudritoSteila}, by proving that $\mathcal…

Cardinal preserving ideals

We give some general criteria, when kappa-complete forcing preserves largeness properties -- like kappa-presaturation of normal ideals on lambda (even when they concentrate on small cofinalities). Then we quite accurately obtain the consistency strength ``NS_lambda is aleph_1-preserving'', for lambda > aleph_2 .