1 - The Computability Concept

Abstract

This chapter concerns computability theory, also known as recursion theory, the area of mathematics dealing with the concept of an effective procedure—a procedure that can be carried out by specific rules. Effective procedures show how limiting the concept of decidability is. One can utilize the concepts of countable and uncountable sets. Computability theory arose before the development of digital computers. It is relevant to certain considerations in mathematical logic. This chapter describes several equivalent ways of formulating the concept in precise terms. The mathematical concept of a computable partial function is the correct formalization of the informal concept of an effectively calculable partial function. It provides a general overview of a number of different ways of formalizing the concept of effective calculability. The idea behind the concept of effective calculable functions is that one should be able to give explicit instructions—a program—for calculating such a function. The precise concept of a computable partial function is an accurate formalization of the informal concept of an effectively calculable function.