We consider implicit definability over the natural number system . We present a new proof of two theorems of Leo Harrington. The first theorem says that there exist implicitly definable subsets of which are not explicitly definable from each other. The second theorem says that there exists a subset of which is not implicitly definable but belongs to a countable, explicitly definable set of subsets of . Previous proofs of these theorems have used finite- or infinite-injury priority constructions. Our new proof is easier in that it uses only a nonpriority oracle construction, adapted from the standard proof of the Friedberg jump theorem.
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