大基数与Chang氏模型C

本文研究了一些基数在Ｃｈａｎｇ氏模型Ｃ中的存在性，证明发如下结果：（１）如果Ｋ是弱紧基数，则Ｋ在Ｃ中也是弱紧基数，（２）如果Ｋ是ｉｎｅｆｆａｂｌｅ基数，则Ｋ在Ｃ中也是ｉｎｅｆｆａｂｌｅ基数。（３）如果Ｋ是完全ｉｎｅｆｆａｂｌｅ基数，则Ｋ在Ｃ中也是完全ｉｎｅｆｆａｂｌｅ基数。（４）设Ｊ：Ｃ→Ｃ为初等嵌入，Ｋ为最小变动的基数，则Ｋ在Ｃ中完全ｉｎｅｆｆａｂｌｅ基数，且是完全Ｒａｍｓｅｙ基数。     

提取合并同位积数法(续)

第三节 基数加减定数法 基数加减定数法中,包括有基数加定法和基数减定法两种,这两种方法,只适应于乘6或乘6以后的数,否则,效果不佳,所以,本节中的基数加定法和基数减定法都从乘6开始。     

Fuzzy幂群的基数定理

文（１）提出了幂群的概念，给出了幂群中各元素是等势的基数定理，文（２）提出了Ｆｕｚｚｙ幂群的概念，但没研究其中各元素的基数问题，本文深入研究这一问题，得到了由Ｄ．Ｄｕｂｏｉｓ等在文（３）中提出的和由李洪兴等在文（４）中提出的两种Ｆｕｚｚｙ集基数形式下的Ｆｕｚｚｙ幂群的基数定理，并给出了Ｆｕｚｚｙ幂群中与基数有关的若干结果。     

提取合并同位积数法（续）

基数加减定数法中，包括有基数加定法和基数减定法两种，这两种方法，只适应于乘6或乘6以后的数，否则，效果不佳，所以，本节中的基数加定法和基数减定法都从乘6开始。     

模糊集的基数与连续统假设

本文在模糊映射的基础上给出了模糊集基数的定义，它把普通集的基数作为特款；不但得到有关基数的大部分结论，而且有其自身的特殊性质；特别，对于连续统假设这一世界难题可能有新的启示．     

强不可接近基数上 P(K)的插入定理

<正> 在符号逻辑杂志1975年6月份第40卷第2期上 Friedman H.选编了102个数理逻辑问题.其中第24个问题是关于无穷正规基数 K 上的命题演算 P(K)是否遵守插入定理的问题.问题后面还引述了 Friedman H.自己的两个结果:当 K 是共尾数为ω的强极限基数的后继基数时,P(K)遵守插入定理.当 K 是一个共尾数大于ω的基数的后     

任意基数集上的拟阵之单扩张

对于由Betten和Wenzel于2003年提出的任意基数集上的拟阵其相应的秩公理给予了证明,并将此结果用于研究任意基数集上的拟阵的单扩张问题.     

Fuzzy映射与F基数

本文定义了从一个Ｆｕｚｚｙ集到另外一个Ｆｕｚｚｙ集的映射，称之为Ｆｕｚｚｙ映射，它不同于以往人们习惯用的“模糊映射”；给出了Ｆｕｚｚｙ映射的等价条件并研究了Ｆｕｚｚｙ映射的性质；基于这样的Ｆｕｚｚｙ映射定义了Ｆｕｚｚｙ映集的基数简称为Ｆ基数，讨论了它的基本性质；最后说明了Ｆ基数对于连续统假设的影响。     

提取合并同位积数法(续)

三、乘8的基数减定法 1、乘8的基数减定法关系表     

链基超滤及其应用

<正> 超积(ultraproduct)作为建立模型的方法已为大家所熟知,但超积模型的基数问题至今仍然知之不多,譬如对于超幂(ultrapower)(?),我们只知道,当(?)是基数为α的集I 上的正规超滤时,有|(?)A|=|A|~a(?)但是否在一致超滤上建立的超幂都具有这种最大可能的基数呢?抑或哪种非正规的一致超滤上建立的超幂也具有最大基数?却不得而知了,与此相关的一个问题是:若 A 是个有序集,譬如是一良序集,它的共尾特征(与 A 共     

A Cardinal Pattern Inspired by AD

Assuming Con(AD), a model in which there are unboundedly many regular cardinals below Θ and in which the only regular cardinals below Θ are limit cardinals was previously constructed. Using a large cardinal hypothesis far beyond Con(AD), we construct in this paper a model in which there is a proper class of regular cardinals and in which the only regular cardinals in the universe are limit cardinals. Mathematics Subject Classification: 03E55, 03E60.     

Superstrong and other large cardinals are never Laver indestructible

Superstrong cardinals are never Laver indestructible. Similarly, almost huge cardinals, huge cardinals, superhuge cardinals, rank-into-rank cardinals, extendible cardinals, 1-extendible cardinals, 0-extendible cardinals, weakly superstrong cardinals, uplifting cardinals, pseudo-uplifting cardinals, superstrongly unfoldable cardinals, Σ_n-reflecting cardinals, Σ_n-correct cardinals and Σ_n-extendible cardinals (all for n ≥  3) are never Laver indestructible. In fact, all these large cardinal properties are superdestructible: if κ exhibits any of them, with corresponding target θ, then in any forcing extension arising from nontrivial strategically <κ-closed forcing \({\mathbb{Q} \in V_\theta}\), the cardinal κ will exhibit none of the large cardinal properties with target θ or larger.     

Gap forcing

In this paper, I generalize the landmark Lévy-Solovay Theorem [LévSol67], which limits the kind of large cardinal embeddings that can exist in a small forcing extension, to a broad new class of forcing notions, a class that includes many of the forcing iterations most commonly found in the large cardinal literature. After such forcing, the fact is that every embedding satisfying a mild closure requirement lifts an embedding from the ground model. Such forcing, consequently, can create no new weakly compact cardinals, measurable cardinals, strong cardinals, Woodin cardinals, strongly compact cardinals, supercompact cardinals, almost huge cardinals, or huge cardinals, and so on. My research has been supported in part by grants from the PSC-CUNY Research Foundation and from the Japan Society for the Promotion of Science. I would like to thank my gracious hosts at Kobe University in Japan for their generous hospitality. This paper follows up an earlier announcement of the main theorem appearing, without technical details, in [Ham99].     

Measurable cardinals and good ‐wellorderings

We study the influence of the existence of large cardinals on the existence of wellorderings of power sets of infinite cardinals κ with the property that the collection of all initial segments of the wellordering is definable by a Σ₁‐formula with parameter κ. A short argument shows that the existence of a measurable cardinal δ implies that such wellorderings do not exist at δ‐inaccessible cardinals of cofinality not equal to δ and their successors. In contrast, our main result shows that these wellorderings exist at all other uncountable cardinals in the minimal model containing a measurable cardinal. In addition, we show that measurability is the smallest large cardinal property that imposes restrictions on the existence of such wellorderings at uncountable cardinals. Finally, we generalise the above result to the minimal model containing two measurable cardinals.     

Tree forcings and sharps

The Levy-Solovay theorem and its variants show that various large cardinals are preserved under small forcings. We study the preservation of consequences of large cardinals under proper forcings. In particular, we consider the preservation of the statement that every real has a sharp for classical tree forcings. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Characterizing strong compactness via strongness

We construct a model in which the strongly compact cardinals can be non‐trivially characterized via the statement “κ is strongly compact iff κ is a measurable limit of strong cardinals”. If our ground model contains large enough cardinals, there will be supercompact cardinals in the universe containing this characterization of the strongly compact cardinals.     

Tallness and level by level equivalence and inequivalence

We construct two models containing exactly one supercompact cardinal in which all non‐supercompact measurable cardinals are strictly taller than they are either strongly compact or supercompact. In the first of these models, level by level equivalence between strong compactness and supercompactness holds. In the other, level by level inequivalence between strong compactness and supercompactness holds. Each universe has only one strongly compact cardinal and contains relatively few large cardinals (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Short extenders forcings – doing without preparations

We introduce certain morass type structures and apply them to blowing up powers of singular cardinals. As a bonus, a forcing for adding clubs with finite conditions to higher cardinals is obtained.     

The least weakly compact cardinal can be unfoldable,weakly measurable and nearly $${\theta}$$-supercompact

We prove from suitable large cardinal hypotheses that the least weakly compact cardinal can be unfoldable, weakly measurable and even nearly \({\theta}\)-supercompact, for any desired \({\theta}\). In addition, we prove several global results showing how the entire class of weakly compactcardinals, a proper class, can be made to coincide with the class of unfoldable cardinals, with the class of weakly measurable cardinals or with the class of nearly \({\theta_\kappa}\)-supercompact cardinals \({\kappa}\), for nearly any desired function \({\kappa\mapsto\theta_\kappa}\). These results answer several questions that had been open in the literature and extend to these large cardinals the identity-crises phenomenon, first identified by Magidor with the strongly compact cardinals.     

Coding into HOD via normal measures with some applications

We develop a new method for coding sets while preserving GCH in the presence of large cardinals, particularly supercompact cardinals. We will use the number of normal measures carried by a measurable cardinal as an oracle, and therefore, in order to code a subset A of κ, we require that our model contain κ many measurable cardinals above κ. Additionally we will describe some of the applications of this result. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim