Mathematical Sciences: Infinite Combinatorics and Applications

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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.