The Foundations of Mathematics: v. 19 (Logic S.)

Mathematical logic grew out of philosophical questions regarding the foundations of mathematics but logic has now outgrown its philosophical roots and has become an integral part of mathematics in general. This book is designed for students who plan to specialize in logic as well as for those who are interested in the applications of logic to other areas of mathematics. Used as a text it could form the basis of a beginning graduate-level course. There are three main chapters: Set Theory Model Theory and Recursion Theory. The Set Theory chapter describes the set-theoretic foundations of all of mathematics based on the ZFC axioms. It also covers technical results about the Axiom of Choice well-orderings and the theory of uncountable cardinals. The Model Theory chapter discusses predicate logic and formal proofs and covers the Completeness Compactness and Löwenheim-Skolem Theorems elementary submodels model completeness and applications to algebra. This chapter also continues the foundational issues begun in the set theory chapter. Mathematics can now be viewed as formal proofs from ZFC. Also model theory leads to models of set theory. This includes a discussion of absoluteness and an analysis of models such as H(κ) and R(γ). The Recursion Theory chapter develops some basic facts about computable functions and uses them to prove a number of results of foundational importance; in particular Churchs theorem on the undecidability of logical consequence the incompleteness theorems of Gödel and Tarskis theorem on the non-definability of truth.