Chapter 2: Gathering Our Tools: Basic Concepts and Notation

In this chapter, we first review a few essential computability theoretic concepts, as a reminder and to fix notation; for more information, see [158,159,170,196,197] or Chapter 2 of [40]. This review assumes knowledge of basic concepts such as computable (also known as recursive) sets and functions, computably enumerable (also known as recursively enumerable) sets, and Turing reductions. We then introduce the important technique of forcing, originally developed in the context of set theory but of great usefulness in computability theory and reverse mathematics.