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A Hierarchy of computably Enumerable Degrees

We introduce a new hierarchy of computably enumerable degrees. This hierarchy is based on computable ordinal notations measuring complexity of approximation of ∆2 functions. The hierarchy unifies and classifies the combinatorics of a number of diverse constructions in computability theory. It does so along the lines of the high degrees (Martin) and the array noncomputable degrees (Downey, Jocksuch and Stob). The hierarchy also gives a number of natural definability results in the c.e. degrees, including a definable