Weakly 2-randoms and 1-generics in scott sets

Research output: Contribution to journalArticle

Abstract

Let S be a Scott set, or even an ω-model of WWKL. Then for each A ∈ S, either there is X ∈ S that is weakly 2-random relative to A, or there is X ∈ S that is 1-generic relative to A. It follows that if A1,⋯, An ∈ S are noncomputable, there is X ∈ 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 inD also holds in DS, where DS is the partial order of Turing degrees of elements of S.

Original languageEnglish (US)
Pages (from-to)392-394
Number of pages3
JournalJournal of Symbolic Logic
Volume83
Issue number1
DOIs
StatePublished - Mar 1 2018

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Partial Order
Turing Degrees
Turing
Language
Model

All Science Journal Classification (ASJC) codes

  • Philosophy
  • Logic

Cite this

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title = "Weakly 2-randoms and 1-generics in scott sets",
abstract = "Let S be a Scott set, or even an ω-model of WWKL. Then for each A ∈ S, either there is X ∈ S that is weakly 2-random relative to A, or there is X ∈ S that is 1-generic relative to A. It follows that if A1,⋯, An ∈ S are noncomputable, there is X ∈ 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 inD also holds in DS, where DS is the partial order of Turing degrees of elements of S.",
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Weakly 2-randoms and 1-generics in scott sets. / Westrick, Linda Brown.

In: Journal of Symbolic Logic, Vol. 83, No. 1, 01.03.2018, p. 392-394.

Research output: Contribution to journalArticle

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AB - Let S be a Scott set, or even an ω-model of WWKL. Then for each A ∈ S, either there is X ∈ S that is weakly 2-random relative to A, or there is X ∈ S that is 1-generic relative to A. It follows that if A1,⋯, An ∈ S are noncomputable, there is X ∈ 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 inD also holds in DS, where DS is the partial order of Turing degrees of elements of S.

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