The Foundations of Mathematics
Kenneth Kunen
College Publications 
<span>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, Church's theorem on the  undecidability of logical consequence, the incompleteness theorems of  Gödel, and Tarski's theorem on the non-definability of truth.</span>