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The typical Turing degree. (English) Zbl 1332.03009
The paper continues the programme initiated by Yates, Martin, Jockusch and Kurtz of understanding the behaviour of “most” Turing degrees in terms of both measure and category. The intuition put forward is that many results which hold of 2-random degrees also hold of all predecessors of 2-random degrees.
The main results proved are:
1.
Every degree bounded by a 2-random degree has a strong minimal cover.
2.
Every degree bounded by a 2-random degree satisfies the join property.
3.
Every degree bounded by a 2-random degree is the join of two 1-generic degrees.
The proofs are an elaboration on Paris’s “measure risking” technique. The common features of these proofs are discussed in a methodological way. We remark that recently L. Bienvenu and L. Patey [“Diagonally non-computable functions and fireworks”, Inf. Comput. (to appear), arXiv:1411.6846] gave another general presentation of the measure-risking method.

MSC:
03D28 Other Turing degree structures
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