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On initial segments of hyperdegrees1

  • S. K. Thomason (a1)

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An initial segment of hyperdegrees is a set S of hyperdegrees such that whenever hS and kh then kS. The main results of this paper affirm the existence of initial segments having certain order types. In particular, if L is a finite distributive lattice then L is isomorphic to an initial segment of hyperdegrees [Theorem 1]; as a consequence the elementary theory of the ordering of hyperdegrees is recursively undecidable [Corollary 1].

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1

This paper reports research supported in part by the National Research Council of Canada, Grant #A-4065.

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[1]Gandy, R. O. and Sacks, G. E., A minimal hyperdegree, Fundamenta mathematicae, vol. 61 (1967), pp. 215223.
[2]Lachlan, A. H., Distributive initial segments of the degrees of unsolvability, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 14 (1968), pp. 457472.
[3]Rosenstein, Joseph G., Initial segments of degrees, Pacific journal of mathematics, vol. 24 (1968), pp. 163172.
[4]Spector, C., On degrees of recursive unsolvability, Annals of mathematics, vol. 64 (1956), pp. 581592.
[5]Thomason, S. K., The forcing method and the upper semi-lattice of hyperdegrees, Transactions of the American Mathematical Society, vol. 129 (1967), pp. 3858.

On initial segments of hyperdegrees1

  • S. K. Thomason (a1)

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