A recursion-theoretic characterization of the ramified analytical hierarchy
Richard Boyd, Gustav B. Hensel, Hilary Putnam · Transactions of the American Mathematical Society · 1969
Step 0: take all the r.e. sets. Step n + 1: add all sets which are r.e. in sets taken at a previous stage. Moreover, this construction is intimately related to the Kleene arithmetical hierarchy, defined in terms of the number and quality of alternating numberquantifiers needed to define a set (using a matrix which is a recursive predicate of integers). In terms of degrees of unsolvability as opposed to sets, what this construction amounts to is: