Much recent research on Turing degrees has focused on the overall structure of the set of Turing degrees and the set of Turing degrees containing recursively enumerable sets.
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This groundbreaking result opened a wide study of the Turing degrees of the recursively enumerable sets which turned out to possess a very complicated and non-trivial structure.
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In computational complexity theory, the complexity class containing all recursively enumerable sets is lattice of r . e . sets under inclusion is denoted \ mathcal { E }.
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But Post left open the main problem of the existence of recursively enumerable sets of intermediate Turing degree; this problem became known as " Post's problem ".
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The theory of first-order Peano arithmetic is consistent, has an infinite but recursively enumerable set of axioms, and can encode enough arithmetic for the hypotheses of the incompleteness theorem.
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If one takes all statements in the language of Peano arithmetic as axioms, then this theory is complete, has a recursively enumerable set of axioms, and can describe addition and multiplication.
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The major research on strong reducibilities has been to compare their theories, both for the class of all recursively enumerable sets as well as for the class of all subsets of the natural numbers.
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In an influential address to the American Mathematical Society in 1944, he raised the question of the existence of an uncomputable recursively enumerable set whose Turing degree is less than that of the halting problem.
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Besides the lattice of recursively enumerable sets, automorphisms are also studied for the structure of the Turing degrees of all sets as well as for the structure of the Turing degrees of r . e . sets.
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The definition of a recursively enumerable set as the " domain " of a partial function, rather than the " range " of a total recursive function, is common in contemporary texts.
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