The Journal of Symbolic Logic

Research Article

Maximal chains in the Turing degrees

C. T. Chonga1 and Liang Yua2 

a1 Department of Mathematics, Faculty of Science, National University of Singapore, Lower Kent Ridge Road, Singapore 117543, Singapore. E-mail: chongct@math.nus.edu.sg

a2 Institute of Mathematical Science, Nanjing University, Nanjing, Jiangsu Province 210093 PR. OF China. E-mail: yuliang.nju@gmail.com

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 ∈ 2 ω , (ω 1) L[a] = ω 1 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”.

(Received June 10 2006)

Footnotes

  The first author wishes to thank Andrea Sorbi and the University of Sienna for their gracious hospitality, during a visit under the INDAM-GNSAGA visiting professorship scheme. His research was also partially supported by NUS grant WBS 146-000-054-123.

  The second author was supported by NUS Grant No. R-146-000-078-112 (Singapore) and NSF of China No, 10471060 and No. 10420130638.