 We have seen that part iv of  Proposition 3.11  characterizes the sets Turing reducible to \(\emptyset'\) as the \(\Delta^0_2\)-definable sets.  Theorem 3.12  thus extends the characterizations of the computable (i.e., \(\Delta^0_1\)-definable) and computably enumerable (i.e., \(\Sigma^0_1\)-definable) sets given in  Proposition 3.8  by demonstrating the coincidence of the \(\Delta^0_2\)-definable sets and those which are limit computable.