Logic: From Foundations to Applications: European Logic Colloquium

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.

Table of Contents:
1: The method of hypersequents in the proof theory of propositional non-classical logic 2: Church-Rosser lambda theories, infinite lambda-terms and consistency problems 3: Baire category for monotone sets 4: Equality in substructural logics 5: Substructural predicates 6: Critical points in an algebra of elementary embeddings II 7: O-minimality nd tame topology 8: Embeddings of Heyting algebras 9: Embedding normal forms 10: Analysing proofs in analysis 11: Subalgebras of Cohen Algebras need not be Cohen 12: Independence structures in set theory 13: Recovering the action of an automorphism group 14: On the logical strength of Nash-Williams' theorem on transfinite sequences 15: Open questions around Büchi and Presburger arithmetics 16: A growth dichotomy for O-minimal expansions of ordered fields 17: Lukasiewicz normal forms and toric desingularizations 18: Fine hierarchy and definability in the Lindenbaum algebra 19: From logic problems to inductive definitions 20: Hyperstable theories 21: Quasi-Riemann Surfaces