Non-cuppable enumeration degrees via finite injury (Q2907057)

The partially ordered local structure \(({\mathcal D}_{\mathrm e}(\leq {\mathbf 0}'_{\mathrm e}),\leq)\), where \({\mathcal D}_{\mathrm e}(\leq {\mathbf 0}'_{\mathrm e})\) is the set of all the \(\Sigma^0_2\) e-degrees and \(\leq\) is the partial ordering induced by the enumeration reducibility \(\leq_{\mathrm e}\), is one of the structures studied in computability theory. In this paper, using finite injury methods, the author strengthens and refines some known results on \(\Sigma^0_2\) e-degrees that have been proved in the literature with priority tree arguments. Namely, it is shown first the existence of an e-high \(\Sigma^0_2\) e-degree incomparable with all the \(\Delta^0_2\) e-degrees; because of the e-highness of the \(\Sigma^0_2\) e-degree, this result refines a special case of a result contained in [\textit{S. B. Cooper} and \textit{C. S. Copestake}, Z. Math. Logik Grundlagen Math. 34, No. 6, 491--522 (1988; Zbl 0667.03034)]. Then the author shows the existence of an e-high noncuppable \(\Sigma^0_2\) e-degree which has the property of being upwards properly \(\Sigma^0_2\).NEWLINENEWLINEFrom the latter there follows the following corollary, which is a result contained in [\textit{S. Bereznyuk, R. Coles} and \textit{A. Sorbi}, J. Symb. Log. 65, No. 1, 19--32 (2000; Zbl 0946.03049)]: For every e-degree \({\mathbf b}\leq{\mathbf 0}'_{\mathrm e}\) there exists an e-degree \({\mathbf c}\) with \({\mathbf b}\leq {\mathbf c}<{\mathbf 0'_{\mathrm e}}\) such that every e-degree \({\mathbf x}\in [{\mathbf c},{\mathbf 0}'_{\mathrm e})\) is properly \(\Sigma^0_2\). Moreover, the author proves the existence of an e-low\(_2\) noncuppable \(\Sigma^0_2\) e-degree. The latter has been previously proved in [\textit{M. Giorgi, A. Sorbi} and \textit{Y. Yang}, J. Symb. Log. 71, No. 4, 1125--1144 (2006; Zbl 1114.03034)]. Finally, combining a finite injury argument with a priority tree argument, it is shown that for every \(\Sigma^0_2\) e-degree \({\mathbf b}\) there exists a noncuppable \(\Sigma^0_2\) e-degree \({\mathbf a}>{\mathbf 0}_{\mathrm e}\) such that \({\mathbf b}'\leq{\mathbf a}''\) and \({\mathbf a}''\leq{\mathbf b}''\). This shows that there are noncuppable \(\Sigma^0_2\) e-degrees at every level of the e-high/e-low hierarchy.