Branching in the enumeration degrees of the \(\Sigma_2^0\) sets
A set \(A\) is enumeration reducible to a set \(B\) (\(A\leq_e B\)) if there exists an effective procedure for enumerating \(A\), given any enumeration of \(B.A\equiv_e B\) iff \(A\leq_e B \& B\leq_e A\). The \(\equiv_e\)-equivalence classes are called enumeration degrees. This paper proves that any incomplete \(\Sigma^0_2\) enumeration degree \(\mathbf c\) is a branching degree, namely, there exist two \(\Sigma^0_2\) enumeration degrees \(\mathbf a\), \(\mathbf b\) such that \(\mathbf c < \mathbf a \& \mathbf c < \mathbf b \& \mathbf c =\mathbf a\wedge \mathbf b\). This shows another elementary difference between enumeration degrees below \({\mathbf{0}}_{\mathbf e}'\) and the recursively enumerable degrees.
scientific article; zbMATH DE number 1310631
scientific article