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a p(e argument change after stage change only finitely characteristic function complete the proof condition construction contradiction cursive Date approved deduction defined and equal denote density theorem enumerating function f and g f(cr fact finite set following theorem hence holds at stage implies International Business Machines J J J J Kleene's L(cr large that cr least member least ordinal least stage least upper bound less Let cr Let f lim f m(cr meta metarecursion theory metarecursively enumerable metadegree metarecursively enumerable sets never holds non-zero notation otherwise partial metarecursive function Post's problem predicate proof of Lemma proof of Theorem range recursion theory recursive ordinals representing function satisfying the theorem set of recursive sive stable stage cr Subcase 1A holds Subcase IB Suppose t(cr Thesis total function transitive relation unbounded set weak relative metarecursiveness weakly metarecursive y(cr