The complex numbers and complex exponentiation: why infinitary logic is necessary!
John T. Baldwin · 2006
In this article we discuss some of the uses of model theory to investigate the structure of the field of complex numbers with exponentiation and associated algebraic groups. After a sketch of some background material on the use of first order model theory in algebra, we describe the inadequacy of the first order framework for studying complex exponentiation. Then, we discuss the Zilber’s program for understanding complex exponentiation using infinitary logic and the essential role of understanding models in cardinality greater than ℵ1. This analysis has inspired a number of algebraic results; we summarize some of them. We close by discussing some consequences on ‘semiabelian varieties’ of the work on the model theory of uncountable models in infinitary logic. We place in context seminal works of Shelah [She75, She83a, She83b] and Zilber [Zil05, Zil00, Zil04, Zil03]. Shelah’s work was directed at understanding model theoretic phenomena–generalizing to infinitary logic the techniques and results that were proving so successful in the first order context. Zilber’s later work was motivated by the attempt to understand complex exponentiation. But he