Weakly 2-randoms and 1-generics in scott sets

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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

All Science Journal Classification (ASJC) codes

  • Philosophy
  • Logic

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