Abstract
The Major Sub-degree Problem of A. H. Lachlan (first posed in 1967) has become a long-standing open question concerning the structure of the computably enumerable (c.e.) degrees. Its solution has important implications for Turing definability and for the ongoing programme of fully characterising the theory of the c.e. Turing degrees. A c.e. degree a is a major subdegree of a c.e. degree b > a if for any c.e. degree x, \({{\bf 0' = b \lor x}}\) if and only if \({{\bf 0' = a \lor x}}\) . In this paper, we show that every c.e. degree b ≠ 0 or 0′ has a major sub-degree, answering Lachlan’s question affirmatively.
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Both authors were funded by EPSRC Research Grant no. GR/M 91419, “Turing Definability”, by INTAS-RFBR Research Grant no. 97-0139, “Computability and Models”, and by an NSFC Grand International Joint Project Grant no. 60310213, “New Directions in Theory and Applications of Models of Computation”. Both authors are grateful to Andrew Lewis for helpful suggestions regarding presentation, technical aspects of the proof, and verification. A. Li is partially supported by National Distinguished Young Investigator Award no. 60325206 (People’s Republic of China).
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Cooper, S.B., Li, A. On Lachlan’s major sub-degree problem. Arch. Math. Logic 47, 341–434 (2008). https://doi.org/10.1007/s00153-008-0083-5
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DOI: https://doi.org/10.1007/s00153-008-0083-5