The \(\text{low}_n\) and \(\text{low}_m\) r.e. degrees are not elementarily equivalent (Q2386567)

The author answers a question raised by \textit{C. G. Jockusch, A. Li} and \textit{Y. Yang} [``A join theorem for the computably enumerable degrees'', Trans. Am. Math. Soc. 356, 2557--2568 (2004; Zbl 1057.03030)]. Jockusch, Li and Yang showed that the low\(_n\) and low\(_1\) r.e. degrees are not elementarily equivalent for \(n>1\). They also asked the question: Are there any \(n\neq m\) such that \(L_n\equiv L_m\)? By making use of the results of \textit{A. Nies, R. A. Shore} and \textit{T. A. Slaman} [``Interpretability and definability in the recursively enumerable degrees'', Proc. Lond. Math. Soc., III. Ser. 77, 241--291 (1998; Zbl 0904.03028)], the author gives a complete answer to the question: For all \(n>m>1\), \(L_n\not\equiv L_m\).