Abstract
AbstractWe 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”.
Publisher
Cambridge University Press (CUP)
Cited by
4 articles.
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1. On Martin’s pointed tree theorem;Computability;2016-05-17
2. Cofinal maximal chains in the Turing degrees;Proceedings of the American Mathematical Society;2014-01-30
3. A Π¹₁-uniformization principle for reals;Transactions of the American Mathematical Society;2009-02-10
4. Thin Maximal Antichains in the Turing Degrees;Lecture Notes in Computer Science;2007