The First Basic Results

Abstract

In the previous chapters we have defined the basic notions and concepts of a theory that we are interested in, Computability Theory. In particular, we have rigorously defined its basic notions, i.e., the notions of algorithm, computation, and computable function. We have also defined some new notions, such as the decidability and semi-decidability of a set, that will play key roles in the next chapter (where we will further develop Computability Theory). As a side product of the previous chapters we have also discovered some surprising facts, such as the existence of the universal Turing machine. It is now time to start using this apparatus and deduce the first theorems of Computability Theory. In this chapter we will first prove several simple but useful theorems about decidable and semi-decidable sets and their relationship. Then we will deduce the so-called Padding Lemma and, based on it, introduce the extremely important concept of the index set. This will enable us to deduce two influential theorems, the Parametrization Theorem and the Recursion Theorem. We will not be excessively formal in our deductions; instead, we will equip them with meaning and motivation wherever appropriate.

Recursion is a method of defining objects in which the object being defined is applied within its own definition.