Mathematical Sciences: Infinite Combinatorics and Applications
数学科学:无限组合及其应用
基本信息
- 批准号:9622579
- 负责人:
- 金额:$ 6万
- 依托单位:
- 依托单位国家:美国
- 项目类别:Continuing Grant
- 财政年份:1996
- 资助国家:美国
- 起止时间:1996-07-01 至 1998-06-30
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
DMS-9622579 PI: Menachem Kojman Carnegie-Mellon University Cantor's discovery of infinite cardinals and his study of their arithmetic gave birth to axiomatic set theory, now accepted as a foundation to mathematics. In spite of the dramatic effect the discovery of infinite cardinals had on mathematics, some of the simplest questions regarding their arithmetic remained unanswered, such as how large the continuum is. The answer finally obtained to this question, by Godel and Cohen, was that the cardinalities of powers of regular cardinals, in particular that of the continuum, are independent of ZFC. The chaotic world-view which emerged after Cohen's and Easton's independence results was challenged some years later by Silver's theorem about powers of singulars, and eventually by Shelah's work on cardinal arithmetic, which restored a dimension of order and regularity to the arithmetic of singular cardinals. The current project employs set theoretic methods, especially those infinite combinatorics discovered in the context of cardinal arithmetic, to study a variety of phenomena in infinite mathematical structures, such as embeddability and homogeneity. Infinite sets come in different "sizes", called cardinals: for example, the whole numbers and the real numbers are both infinite but the set of real numbers has the greater cardinal. Infinite cardinals have their own arithmetic; one can add, multiply and take powers, as with ordinary arithmetic. Research in the arithmetic of cardinals led to the development of Axiomatic Set Theory, which in turn has provided a foundations for mathematics. There are two kinds of infinite cardinals, Regular and Singular. The arithmetic of Regular cardinals was studied first and shown to be chaotic. The arithmetic of Singular cardinals turns out to be better behaved. In recent years a coherent theory has developed, called "pcf theory" by its inventor, Saharon Shelah, bringing some order to the realm of cardinal arithmetic, order which was conspicuous in its absence for almost a century. This project employs pcf theory to discover patterns and interrelations among other infinite mathematical structures, in order to classify these structures according to complexity, generality, and various symmetry properties.
DMS-9622579 PI:卡内基-梅隆大学康托发现无限基数并研究它们的算术,催生了公理集合论,现在被认为是数学的基础。尽管无限基数的发现对数学产生了戏剧性的影响,但关于他们的算术的一些最简单的问题仍然没有得到回答,比如连续体有多大。对于这个问题,哥德尔和科恩最终得到的答案是,正则基数的幂的基数,特别是连续统的幂的基数,是独立于ZFC的。在科恩和伊斯顿的独立性结果之后出现的混乱世界观,几年后受到了希尔弗关于奇点幂的定理的挑战,最终受到了谢拉关于基数算术的工作的挑战,该工作将秩序和规律性的维度恢复到了奇数基数的算术中。本课题采用集合论方法,特别是在基数算术背景下发现的无限组合数学方法,来研究无限数学结构中的各种现象,如可嵌入性和齐性。无限集合有不同的“大小”,称为基数:例如,整数和实数都是无限的,但实数集具有更大的基数。无限的基数有他们自己的算术;人们可以像普通算术一样加、乘和取幂。对基数算术的研究导致了公理集合论的发展,而公理集合论又为数学奠定了基础。无限基数有两种,正则基数和奇数基数。首先研究了正则基数的运算,证明了其具有混沌特性。事实证明,奇数基数的运算表现得更好。近年来,一种连贯的理论已经发展起来,它的发明者撒哈拉·谢拉将其称为“PCF理论”,给基数算术领域带来了一些秩序,秩序在近一个世纪的时间里一直没有出现,这一秩序是引人注目的。这个项目使用PCF理论来发现其他无限数学结构之间的模式和相互关系,以便根据复杂性、一般性和各种对称性对这些结构进行分类。
项目成果
期刊论文数量(0)
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Menachem Kojman其他文献
A symmetrized metric ramsey theorem
- DOI:
10.1007/s11856-009-0093-y - 发表时间:
2009-12-04 - 期刊:
- 影响因子:0.800
- 作者:
Menachem Kojman - 通讯作者:
Menachem Kojman
On the arithmetic of density
- DOI:
10.1016/j.topol.2016.08.016 - 发表时间:
2016-11-01 - 期刊:
- 影响因子:
- 作者:
Menachem Kojman - 通讯作者:
Menachem Kojman
Noetherian type in topological products
- DOI:
10.1007/s11856-014-1101-4 - 发表时间:
2014-07-23 - 期刊:
- 影响因子:0.800
- 作者:
Menachem Kojman;David Milovich;Santi Spadaro - 通讯作者:
Santi Spadaro
Ramsey theory over partitions II: Negative Ramsey relations and pump-up theorems
- DOI:
10.1007/s11856-023-2574-9 - 发表时间:
2023-11-13 - 期刊:
- 影响因子:0.800
- 作者:
Menachem Kojman;Assaf Rinot;Juris Steprāns - 通讯作者:
Juris Steprāns
Cantor-Bendixson degrees and convexity in ℝ2
- DOI:
10.1007/bf02802497 - 发表时间:
2001-12-01 - 期刊:
- 影响因子:0.800
- 作者:
Menachem Kojman - 通讯作者:
Menachem Kojman
Menachem Kojman的其他文献
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