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A theorem on initial segments of degrees1

  • S. K. Thomason (a1)

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A set S of degrees is said to be an initial segment if cdS→-cS. Shoenfield has shown that if P is the lattice of all subsets of a finite set then there is an initial segment of degrees isomorphic to P. Rosenstein [2] (independently) proved the same to hold of the lattice of all finite subsets of a countable set. We shall show that “countable set” may be replaced by “set of cardinality at most that of the continuum.” This result is also an extension of [3, Corollary 2 to Theorem 15], which states that there is a sublattice of degrees isomorphic to the lattice of all finite subsets of 2N. (A sublattice of degrees is a subset closed under ∪ and ∩; an initial segment closed under ∪ is necessarily a sublattice, but not conversely.)

It seems worth noting that the proof of the present result was preceded in time by our proof [4] of the analogous theorem for hyperdegrees, and is in fact an adaptation of that proof. Thus the present work has been influenced much more directly by the Gandy-Sacks forcing construction of a minimal hyperdegree [1] than by previous work on initial segments of degrees.

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1

This work was supported in part by the National Research Council of Canada, grant number 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]Rosenstein, Joseph G., Initial segments of degrees, Pacific journal of mathematics, vol. 24 (1968), pp. 163172.
[3]Thomason, S. K., The forcing method and the upper semilattice of hyperdegrees, Transactions of the American Mathematical Society, vol. 129 (1967), pp. 3857.
[4]Thomason, S. K., On initial segments of hyperdegrees (to appear).

A theorem on initial segments of degrees1

  • S. K. Thomason (a1)

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