The completeness of arithmetic sets under operations of set theory (Q1902753)

In his report on Keele Conference on Mathematical Logic (Keele, England, July 20-29, 1993), D. H. J. de Jongh formulated the following problem: Let \(A \subseteq \omega\) be a \(\Sigma^0_2\)-complete set, \(B\) a \(\Pi^0_2\)-complete subset of \(A\). Is the difference \(A - B\) \(\Sigma^0_2\)-complete? Relying on our criterion of \(\Sigma^0_2\)-completeness, in Theorem 2 we prove a more general result which implies that the answer to de Jongh's question is positive even if we have the completeness condition taking place on an appropriate level of the arithmetical hierarchy only in one of the sets (either \(A\) or \(B)\). In Theorem 3 we prove a similar result for \(\Pi^0_3\)-sets.