This is the Volume II of a text for a problem-oriented undergraduate course in mathematical logic. It covers the basics of computability, using Turing machines and recursive functions, and Goedel's Incompleteness Theorem, and could be used for a one semester course on these topics. Volume I, Propositional and First-Order Logic, covers the basics of these topics through the Soundness, Completeness, and Compactness Theorems. Information on availability and the conditions under which this book may be used and reproduced are given in the preface.