Extensions of two constructions of Ahmad

Abstract

In her 1990 thesis, Ahmad showed that there is a so-called “Ahmad pair”, i.e., there are incomparable ${\mathrm{\Sigma }}_2^0$ -enumeration degrees  ${\mathbf{a}}_0$ and  ${\mathbf{a}}_1$ such that every enumeration degree $\mathbf{x}\mathrm{<}{\mathbf{a}}_0$ is $⩽{\mathbf{a}}_1$ . At the same time, she also showed that there is no “symmetric Ahmad pair”, i.e., there are no incomparable ${\mathrm{\Sigma }}_2^0$ -enumeration degrees  ${\mathbf{a}}_0$ and  ${\mathbf{a}}_1$ such that every enumeration degree ${\mathbf{x}}_0\mathrm{<}{\mathbf{a}}_0$ is $⩽{\mathbf{a}}_1$ and such that every enumeration degree ${\mathbf{x}}_1\mathrm{<}{\mathbf{a}}_1$ is $⩽{\mathbf{a}}_0$ .

In this paper, we first present a direct proof of Ahmad’s second result. We then show that her first result cannot be extended to an “Ahmad triple”, i.e., there are no ${\mathrm{\Sigma }}_2^0$ -enumeration degrees  ${\mathbf{a}}_0$ , ${\mathbf{a}}_1$ and  ${\mathbf{a}}_2$ such that both $\mathrm{(}{\mathbf{a}}_0\mathrm{,}{\mathbf{a}}_1\mathrm{)}$ and $\mathrm{(}{\mathbf{a}}_1\mathrm{,}{\mathbf{a}}_2\mathrm{)}$ are an Ahmad pair. On the other hand, there is a “weak Ahmad triple”, i.e., there are pairwise incomparable ${\mathrm{\Sigma }}_2^0$ -enumeration degrees  ${\mathbf{a}}_0$ , ${\mathbf{a}}_1$ and  ${\mathbf{a}}_2$ such that every enumeration degree $\mathbf{x}\mathrm{<}{\mathbf{a}}_0$ is also $⩽{\mathbf{a}}_1$ or $⩽{\mathbf{a}}_2$ ; however neither $\mathrm{(}{\mathbf{a}}_0\mathrm{,}{\mathbf{a}}_1\mathrm{)}$ nor $\mathrm{(}{\mathbf{a}}_0\mathrm{,}{\mathbf{a}}_2\mathrm{)}$ is an Ahmad pair.