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Computability theory is a branch of mathematical logic and computer science that has become increasingly relevant in recent years. The field has developed growing connections in diverse areas of mathematics, with applications in topology, group theory, and other subfields. This book introduces a new hierarchy that allows them to classify the combinatorics of constructions from many areas of computability theory, including algorithmic randomness, Turing degrees, effectively closed sets, and effective structure theory. This unifying hierarchy gives rise to new natural definability results for Turing degree classes, demonstrating how dynamic constructions become reflected in definability. The book presents numerous construction techniques involving high-level nonuniform arguments, and their self-contained work is appropriate for graduate students and researchers.

Unsolvability (Mathematical logic) --- Computable functions. --- Recursively enumerable sets. --- Enumerable sets, Recursively --- Sets, Recursively enumerable --- Recursion theory --- Computability theory --- Functions, Computable --- Partial recursive functions --- Recursive functions, Partial --- Constructive mathematics --- Decidability (Mathematical logic) --- Degrees of unsolvability --- Turing degrees of unsolvability --- Recursive functions --- Recursion theory. --- c.e. degrees. --- c.e. reals. --- computable model theory. --- lattice embeddings. --- m-topped degrees. --- mind changes in computability theory. --- modern computability theory. --- pi-zero-one classes. --- prompt permissions. --- relative recursive randomness. --- transfinite hierarchy of Turing degrees.