Mathematical Sciences: Infinite Combinatorics and Applications

DMS-9622579 PI：Menachem Kojman梅吉-梅隆大学 康托的发现无限基数和他的研究，他们的算术诞生了公理集理论，现在接受作为基础的数学。尽管无穷基数的发现对数学产生了巨大的影响，但关于它们的算术的一些最简单的问题仍然没有答案，例如连续统有多大。 哥德尔和科恩对这个问题的最终回答是，正则基数的幂的基数，特别是连续统的幂的基数，独立于ZFC。科恩和伊斯顿的独立结果之后出现的混沌世界观在几年后受到了银关于奇点幂的定理的挑战，并最终受到了谢拉关于基数算术的工作的挑战，谢拉的工作恢复了奇异基数算术的有序性和规律性。 目前的项目采用集合论的方法，特别是那些在基数算术的背景下发现的无限组合学，来研究各种 无限数学结构中的现象，如可嵌入性和同质性。 无限集合有不同的“大小”，称为基数：例如，整数和真实的数都是无限的，但真实的数的集合具有更大的基数。 无限枢机有自己的算术;一个可以加，乘和采取权力，与普通算术。对基数运算的研究导致了公理集的发展 理论，这反过来又为数学提供了基础。 有两种无限基数，正则和奇异。首先研究了正则基数的算法，证明了正则基数的算法是混沌的。 奇异基数变换的算法 是为了表现得更好 近年来，一个连贯的理论已经发展起来，称为 “pcf理论”由其发明者，撒哈拉谢拉，带来了一些秩序的领域，基数算术，秩序是引人注目的，在其缺席了近世纪。本计画利用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、通用性和各种对称性来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.