A Recursion-theoretic Characterization of the Ramified Analytical Hierarchy
نویسندگان
چکیده
Introduction. According to a classical theorem of Post, the arithmetical sets may be obtained by the following construction : 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 : Step n («=0, 1, 2,...): "take" all sets recursively enumerable in sets of the degree /n)(0). [Here j denotes the ordinary jump operator on degrees, /0)(0)=0, /n+1)(0) =A/<")(0)).] If r is a collection of degrees, we write RU(F) for the collection of all sets recursive in sets whose degrees are in I\ RU(T) is the smallest class of sets containing all sets whose degrees are in F and closed under "recursive in". The arithmetical hierarchy is thus represented by the linear sequence of degrees 0, j(0),jj(Q),.. -,jM(0),... in the sense that the successive collections
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