### 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 A_{1},⋯, A_{n} ∈ S are noncomputable, there is X ∈ S such that each A_{i} 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 D^{S}, where D^{S} is the partial order of Turing degrees of elements of S.

Original language | English (US) |
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Pages (from-to) | 392-394 |

Number of pages | 3 |

Journal | Journal of Symbolic Logic |

Volume | 83 |

Issue number | 1 |

DOIs | |

State | Published - Mar 1 2018 |

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### All Science Journal Classification (ASJC) codes

- Philosophy
- Logic

### Cite this

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*Journal of Symbolic Logic*, vol. 83, no. 1, pp. 392-394. https://doi.org/10.1017/jsl.2017.73

**Weakly 2-randoms and 1-generics in scott sets.** / Westrick, Linda Brown.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Weakly 2-randoms and 1-generics in scott sets

AU - Westrick, Linda Brown

PY - 2018/3/1

Y1 - 2018/3/1

N2 - 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.

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.

UR - http://www.scopus.com/inward/record.url?scp=85046342760&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85046342760&partnerID=8YFLogxK

U2 - 10.1017/jsl.2017.73

DO - 10.1017/jsl.2017.73

M3 - Article

AN - SCOPUS:85046342760

VL - 83

SP - 392

EP - 394

JO - Journal of Symbolic Logic

JF - Journal of Symbolic Logic

SN - 0022-4812

IS - 1

ER -