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WEAKLY 2-RANDOMS AND 1-GENERICS IN SCOTT SETS

  • LINDA BROWN WESTRICK (a1)
Abstract

Let ${\cal S}$ be a Scott set, or even an ω-model of WWKL. Then for each A ε S, either there is X ε S that is weakly 2-random relative to A, or there is X ε S that is 1-generic relative to A. It follows that if A1,…,An ε S are noncomputable, there is X ε S such that each Ai is Turing incomparable with X, answering a question of Kučera and Slaman. More generally, any ∀∃ sentence in the language of partial orders that holds in ${\cal D}$ also holds in ${{\cal D}^{\cal S}}$ , where ${{\cal D}^{\cal S}}$ is the partial order of Turing degrees of elements of ${\cal S}$ .

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[1]Conidis, C. J., A measure-theoretic proof of Turing incomparability. Annals of Pure and Applied Logic, vol. 162 (2010), no. 1, pp. 8388.
[2]>Downey, R., Nies, A., Weber, R., and Yu, L., Lowness and nullsets, this Journal, vol. 71 (2006), no. 3, pp. 10441052.
[3]Kučera, A. and Slaman, T. A., Turing incomparability in Scott sets. Proceedings of the American Mathematical Society, vol. 135 (2007), no. 11, pp. 37233731.
[4]>Lerman, M., Degrees of Unsolvability, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1983.
[5]>Li, W. and Slaman, T. A., Private communication.
[6]>Nies, A., Computability and Randomness, Oxford Logic Guides, vol. 51, Oxford University Press, Oxford, 2009.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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