a1 Department of Mathematics, Faculty of Science, National University of Singapore, Lower Kent Ridge Road, Singapore 117543, Singapore. E-mail: email@example.com
a2 Institute of Mathematical Science, Nanjing University, Nanjing, Jiangsu Province 210093 PR. OF China. E-mail: firstname.lastname@example.org
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)
† 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.