Implicit Definability in Arithmetic

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