Highness, locally noncappability and nonboundings (Q1944327)

By degree we mean r.e. degree throughout. Call the degree \(\mathbf a\) locally noncappable if there is a degree \(\mathbf c > \mathbf a\) such that for no nonzero \(\mathbf w < \mathbf c\) is \(\mathbf a \wedge \mathbf w = \mathbf 0\). Here \(\mathbf c\) is a witness that \(\mathbf a\) is locally noncappable. The paper's main theorem shows that if \(\mathbf 0 < \mathbf a < \mathbf 0'\) then there is a \(\text{high}_2\) degree \(\mathbf c \succ \mathbf a\) that is such a witness. This improves Seetapun's result, which established only local noncappability. The theorem has as corollaries the known results about \(\text{high}_2\) nonboundings, including the existence of a \(\text{high}_2\) degree bounding no minimal pairs [\textit{R. G. Downey} et al., Math. Log. Q. 39, No. 4, 475--491 (1993; Zbl 0809.03029)] and the continuity result that for any minimal pair \(\{\mathbf a,\mathbf b\}\) there is another minimal pair \(\{\mathbf a,\mathbf c\}\) with \(\mathbf b < \mathbf c\) and \(\mathbf c\) \(\text{high}_2\) [\textit{L. Harrington} and \textit{R. I. Soare}, Math. Sci. Res. Inst. Publ. 26, 39--62 (1992; Zbl 0781.03031)]. A degree \(\mathbf a > \mathbf 0\) is a Harrington plus-cupping degree if for every \(\mathbf b\) and \(\mathbf c\) with \(\mathbf 0 < \mathbf b < \mathbf a\) and \(\mathbf a < \mathbf c\) there exists a degree \(\mathbf d < \mathbf c\) with \(\mathbf b \vee \mathbf d = \mathbf c\). The theorem yields the corollary that above any Harrington plus-cupping degree there is another such degree that is \(\text{high}_2\). There are similar consequences for Fejer-Soare plus-cupping degrees and bases for Slaman triples. A theorem scheme is derived. For any property \(P\) of degrees, say that \(\mathbf a\) is plus-\(P\) if every nonzero degree below \(\mathbf a\) has \(P\). Then if \(P\) is closed upwards and \(\mathbf a\) is plus-\(P\), there is a \(\text{high}_2\) degree above \(\mathbf a\) that is also plus-\(P\).