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Maximal chains in the Turing degrees

  • C. T. Chong (a1) and Liang Yu (a2)

We study the problem of existence of maximal chains in the Turing degrees. We show that:

1. ZF + DC + “There exists no maximal chain in the Turing degrees” is equiconsistent with ZFC + “There exists an inaccessible cardinal”

2. For all a ∈ 2ω, (ω1)L[a] = ω1 if and only if there exists a [a] maximal chain in the Turing degrees. As a corollary, ZFC + “There exists an inaccessible cardinal” is equiconsistent with ZFC + “There is no (bold face) maximal chain of Turing degrees”.

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[17] Robert M. Solovay , A model of set-theory in which every set of reals is Lebesgue measurable. Annals of Mathematics (2), vol. 92 (1970), pp. 156.

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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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