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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)


A $ \Pi^1_1$-uniformization principle for reals

Authors: C. T. Chong and Liang Yu
Journal: Trans. Amer. Math. Soc. 361 (2009), 4233-4245
MSC (2000): Primary 03D28, 03E35, 28A20
Published electronically: February 10, 2009
MathSciNet review: 2500887
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Abstract: We introduce a $ \Pi^1_1$-uniformization principle and establish its equivalence with the set-theoretic hypothesis $ (\omega_1)^L=\omega_1$. This principle is then applied to derive the equivalence, to suitable set-theoretic hypotheses, of the existence of $ \Pi^1_1$-maximal chains and thin maximal antichains in the Turing degrees. We also use the $ \Pi^1_1$-uniformization principle to study Martin's conjectures on cones of Turing degrees, and show that under $ V=L$ the conjectures fail for uniformly degree invariant $ \Pi^1_1$ functions.

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Additional Information

C. T. Chong
Affiliation: Department of Mathematics, Faculty of Science, National University of Singapore, Lower Kent Ridge Road, Singapore 117543

Liang Yu
Affiliation: Institute of Mathematical Sciences, Nanjing University, Nanjing, Jiangsu Province 210093, People’s Republic of China

PII: S 0002-9947(09)04783-7
Received by editor(s): August 14, 2007
Published electronically: February 10, 2009
Additional Notes: The research of the first author was supported in part by NUS grant WBS 146-000-054-123
The second author was supported by NSF of China Grant No. 10701041, Research Fund for Doctoral Programs of Higher Education, No. 20070284043, and Scientific Research Foundation for Returned Overseas Chinese Scholars.
Article copyright: © Copyright 2009 American Mathematical Society

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