Mathematical Sciences: Infinite Combinatorics and Applications

卡耐基梅隆大学卡耐基-梅隆大学康托尔对无限基数的发现和他对无限基数算术的研究产生了公理集合论，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巨大的影响，但一些最简单的算术问题仍然没有得到解答，比如连续统有多大。哥德尔和科恩最终得到了这个问题的答案，即正则基数的幂的基数，特别是连续统的基数，与ZFC无关。在科恩和伊斯顿的独立结果之后出现的混乱世界观在几年后被西尔弗关于奇点幂的定理所挑战，最终被希拉在基数算术方面的工作所挑战，该工作恢复了奇异基数算术的秩序和规律性。目前的项目采用集合论方法，特别是在基数算术的背景下发现的无限组合学，来研究无限数学结构中的各种现象，如可嵌入性和同质性。无限集有不同的“大小”，称为基数：例如，整数和实数都是无限的，但实数集的基数更大。无限基数有自己的算法；我们可以像做普通算术一样，加，乘，乘方。对基数算术的研究导致了公理化集理论的发展，而公理化集理论又为数学奠定了基础。无限基数有正则和奇异两种。首先研究了规则基数的算法，并证明了它是混沌的。奇异基数的算法表现得更好。近年来，一种被其发明者萨哈伦·希拉称为“pcf理论”的连贯理论发展起来，给基数算术领域带来了一些秩序，这种秩序在近一个世纪的缺失中是显而易见的。本项目利用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。

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.