×

Finite cupping sets. (English) Zbl 1060.03067

The author shows that, given any Turing degree \(0<c\leq 0^{\prime }\) and any uniformly \(\Delta _{2}\) sequence of degrees \(b_{0},b_{1},b_{2},\dots\) such that \(\forall i\) \(( b_{i}\nsucceq c) \), there exists \(0<a<0^\prime\) such that for all \(i\geq 0\), \(a\vee b_{i}\nsucceq c\). Then, by considering the case \(c=0^{\prime }\), he concludes that there exists no ‘finite cupping set’, i.e. that there exists no finite set of degree strictly below \(0^{\prime },b_{0},b_{1},\dots ,b_{l}\), say, such that for every degree \(0<a<0^{\prime }\) there exists \(0\leq i\leq l\), \(0^{\prime }=a\vee b_{i}. \)

MSC:

03D28 Other Turing degree structures
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Lewis, A.: Minimal Complements For Degrees Below 0’, Jan 02, unpublished · Zbl 1086.03031
[2] Lewis, A.: A Single Minimal Complement For The C.E. Degrees, Apr 02, unpublished · Zbl 1128.03032
[3] Li, A., Yi, X.: Cupping the recursively enumerable degrees by d.r.e. degrees, Proc. London Math. Soc. 79 (3) , 1–21 (1999) · Zbl 1025.03032 · doi:10.1112/S0024611599011818
[4] Posner, Robinson, Degrees Joining to 0’, 1981 J. Symbolic Logic 46, 714–722 (1981) · Zbl 0517.03014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.