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Degrees of members of π 0 1 classes

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  1. Tez No: 401603
  2. Yazar: AHMET ÇEVİK
  3. Danışmanlar: DR. ANDY LEWIS
  4. Tez Türü: Doktora
  5. Konular: Matematik, Mathematics
  6. Anahtar Kelimeler: Belirtilmemiş.
  7. Yıl: 2014
  8. Dil: İngilizce
  9. Üniversite: University of Leeds
  10. Enstitü: Yurtdışı Enstitü
  11. Ana Bilim Dalı: Belirtilmemiş.
  12. Bilim Dalı: Belirtilmemiş.
  13. Sayfa Sayısı: 116

Özet

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Özet (Çeviri)

In this thesis we study Turing degrees of members of 0 1 classes. We give two introductory chapters and then three main chapters which include new results. In the first chapter we give some standard background for recursion theory, and we give an introduction to 0 1 classes in the second chapter. The third chapter will be on the published work [1]. We show that for any degree a ≥ 0′, if a 0 1 class P contains members of every degree b such that b′ = a, then P contains members of every degree. A local version of this result is also given. That is, when a is also 0 2 , it suffices in the hypothesis to have a member of every 0 2 degree b such that b′ = a. This result extends the Low Antibasis Theorem given in Kent and Lewis [2]. The fourth chapter has three subsections. The first subsection concerns an observation, which may be seen as a cupping non-basis analogue of Jockusch and Soare's capping basis theorem: We show that there exists a non-empty 0 1 class with no recursive member, such that no join of two sets in the class computes ∅′. The second one contains the principal result of the chapter, which concerns the relationship between the join property and the members of 0 1 classes. We show that there exists a non-empty 0 1 class with no recursive member, for which it also holds that no member satisfies the join property. Third subsection contains some future work where we give some open questions about the relation between minimal covers and 0 1 classes, and also about the relation between minimal covers and PA degrees. In the fifth chapter we study the degree spectrum properties of a special kind of 0 1 classes that we introduce, so called 0 1 choice classes. A 0 1 choice class is a 0 1 class such that no two members have the same Turing degree.Considering this restricted class leads us to some interesting antibasis theorems and technically innovative constructions.

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