Abstract

Computability theory is the mathematical theory of algorithms, which explores the power and limitations of computation. Classical computability theory formalized the intuitive notion of an algorithm and provided a theoretical basis for digital computers. It also demonstrated the limitations of algorithms and showed that most sets of natural numbers and the problems they encode are not decidable (Turing computable). Important results of modern computability theory include the classification of the computational difficulty of sets and problems. Arithmetical and hyperarithmetical hierarchies and the Turing degrees provide important measures of the level of such difficulty. There are uncountably many Turing degrees and they are partially ordered. These classifications are used in contemporary research to investigate the complexity of problems in modern computable mathematics, for example, regarding structures and their properties.