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Maximal chains in the Turing degrees

Published online by Cambridge University Press:  12 March 2014

C. T. Chong  and
Liang Yu
C. T. Chong
Affiliation:
Department of Mathematics, Faculty of Science, National University of Singapore, Lower Kent Ridge Road, Singapore 117543, Singapore. E-mail: chongct@math.nus.edu.sg
Liang Yu
Affiliation:
Institute of Mathematical Science, Nanjing University, Nanjing, Jiangsu Province 210093 PR. OF China. E-mail: yuliang.nju@gmail.com
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Abstract

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”.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 72 , Issue 4 , December 2007 , pp. 1219 - 1227
Copyright
Copyright © Association for Symbolic Logic 2007

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