ON THE COMPLEXITY OF THE SUCCESSIVITY RELATION IN COMPUTABLE LINEAR ORDERINGS
2010 ◽
Vol 10
(01n02)
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pp. 83-99
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Keyword(s):
In this paper, we solve a long-standing open question (see, e.g. Downey [6, Sec. 7] and Downey and Moses [11]), about the spectrum of the successivity relation on a computable linear ordering. We show that if a computable linear ordering [Formula: see text] has infinitely many successivities, then the spectrum of the successivity relation is closed upwards in the computably enumerable Turing degrees. To do this, we use a new method of constructing [Formula: see text]-isomorphisms, which has already found other applications such as Downey, Kastermans and Lempp [9] and is of independent interest. It would seem to promise many further applications.