3. Naming Systems

So far we have defined computability on the sets Σ* of finite words and Σω of infinite sequences explicitly by Type-2 machines (Sect. 2.1). In TTE we introduce computability on other sets M by using finite or infinite words as “names”. Machines, therefore, still transform “concrete” sequences of symbols. Only the user of the machine interprets theses sequences as finite or infinite names of “abstract objects”. Although there are several other suggestions to define computability on sets or structures, in this book we will confine ourselves exclusively to computability concepts induced by naming systems. As we have seen, some concepts from computability theory have formally similar topological counterparts. In the following we will continue to develop computability theory, considering also the weaker topological aspects whenever advisable.