## Notre Dame Journal of Formal Logic

### Superhighness

#### Abstract

We prove that superhigh sets can be jump traceable, answering a question of Cole and Simpson. On the other hand, we show that such sets cannot be weakly 2-random. We also study the class $superhigh^\diamond$ and show that it contains some, but not all, of the noncomputable K-trivial sets.

#### Article information

Source
Notre Dame J. Formal Logic Volume 50, Number 4 (2009), 445-452.

Dates
First available in Project Euclid: 11 February 2010

http://projecteuclid.org/euclid.ndjfl/1265899124

Digital Object Identifier
doi:10.1215/00294527-2009-020

Zentralblatt MATH identifier
05778810

Mathematical Reviews number (MathSciNet)
MR2598873

Subjects
Primary: 03D28
Secondary: 03D32 68Q30

#### Citation

Kjos-Hanssen , Bjørn; Nies , Andrée. Superhighness. Notre Dame Journal of Formal Logic 50 (2009), no. 4, 445--452. doi:10.1215/00294527-2009-020. http://projecteuclid.org/euclid.ndjfl/1265899124.

#### References

• [1] Barmpalias, G., "Tracing and domination in the Turing degrees". forthcoming.
• [2] Binns, S., B. Kjos-Hanssen, M. Lerman, and R. Solomon, "On a conjecture of Dobrinen and Simpson concerning almost everywhere domination", The Journal of Symbolic Logic, vol. 71 (2006), pp. 119--36.
• [3] Cholak, P., N. Greenberg, and J. S. Miller, "Uniform almost everywhere domination", The Journal of Symbolic Logic, vol. 71 (2006), pp. 1057--72.
• [4] Cole, J. A., and S. G. Simpson, "Mass problems and hyperarithmeticity", Journal of Mathematical Logic, vol. 7 (2007), pp. 125--43.
• [5] Cooper, S. B., "Minimal degrees and the jump operator", The Journal of Symbolic Logic, vol. 38 (1973), pp. 249--71.
• [6] Dobrinen, N. L., and S. G. Simpson, "Almost everywhere domination", The Journal of Symbolic Logic, vol. 69 (2004), pp. 914--22.
• [7] Friedberg, R., "A criterion for completeness of degrees of unsolvability", The Journal of Symbolic Logic, vol. 22 (1957), pp. 159--60.
• [8] Jockusch, C. G., Jr., and R. I. Soare, "$\Pi \sp{0}\sb{1}$ classes and degrees of theories", Transactions of the American Mathematical Society, vol. 173 (1972), pp. 33--56.
• [9] Martin, D. A., "Classes of recursively enumerable sets and degrees of unsolvability", Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 12 (1966), pp. 295--310.
• [10] Mohrherr, J., "Density of a final segment of the truth-table degrees", Pacific Journal of Mathematics, vol. 115 (1984), pp. 409--19.
• [11] Nies, A., "Superhighness and strong jump traceability". forthcoming in Proceedings of the ICALP 2009.
• [12] Nies, A., Computability and Randomness, Oxford University Press, Oxford, 2009.
• [13] Schwarz, S., Index Sets of Computably Enumerable Sets, Quotient Lattices, and Computable Linear Orderings, Ph.D. thesis, University of Chicago, Chicago, 1982.
• [14] Simpson, S. G., "Almost everywhere domination and superhighness", Mathematical Logic Quarterly, vol. 53 (2007), pp. 462--82.