each formula Wi or 9 being obtained by juxtaposition from atomic symbols and variables ranging over strings of atomic symbols. A set a of strings is r.e. if and only if there is a finite system of such rules and an atomic symbol a such that a string X belongs to a if and only if the string aX is derivable. In the following it will be shown that a similar possibility is available for hyperarithmetical relations. Just as recursive sets may be characterized as sets a such that a and its complement oc' are definable by finitely many rules of the form (1), so we shall show that hyperarithmetical sets may be characterized as sets oc such that oc and oc' are definable by finitely many rules of a certain more general form which we shall specify. Without loss of generality we may restrict ourselves to sets of (natural) numbers 0, 0 O'0, A set oc of numbers is hyperarithmetical if and only if the sets a and oa' are represented in the analytic hierarchy of Kleene [9] by formulas with one universal function-quantifier
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