Logic: From Foundations to Applications

This book contains twenty-one essays by leading authorities on aspects of contemporary logic, ranging from foundations of set theory to applications of logic in computing and in the theory of fields. In those parts of logic closest to computer science, the gap between foundations and applications is often small, as illustrated by three essays on the proof theory of non-classical logics. There are also chapters on the lambda calculus, on relating logic programs to inductive definitions, on Buechi and Presburger arithmetics, and on definability in Lindenbaum algebras. Aspects of constructive mathematics discussed are embeddings of Heyting algebras and proofs in mathematical anslysis. Set theory is well covered with six chapters discussing Cohen forcing, Baire category, determinancy, Nash-Williams theory, critical points (and the remarkable connection between them and properties of left distributive operations) and independent structures. The longest chapter in the book is a survey of 0-minimal structures, by Lou van den Dries; during the last ten years these structures have come to take a central place in applications of model theory to fields and function theory, and this chapter is the first broad survey of the area. Other chapters illustrate how to apply model theory to field theory, complex geometry and groups, and how to recover from its automorphism group. Finally, one chapter applies to the theory of toric varieties to solve problems about many-valued logics.