Transactions of the American Mathematical Society

Author(s): C. T. Chong; Liang Yu
Journal: Trans. Amer. Math. Soc. 361 (2009), 4233-4245.
MSC (2000): Primary 03D28, 03E35, 28A20
Posted: February 10, 2009
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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

the conjectures fail for uniformly degree invariant $\Pi^1_1$ functions.

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C. T. Chong
Affiliation: Department of Mathematics, Faculty of Science, National University of Singapore, Lower Kent Ridge Road, Singapore 117543
Email: chongct@math.nus.eud.sg

Liang Yu
Affiliation: Institute of Mathematical Sciences, Nanjing University, Nanjing, Jiangsu Province 210093, People's Republic of China
Email: yuliang.nju@gmail.com

DOI: 10.1090/S0002-9947-09-04783-7
PII: S 0002-9947(09)04783-7
Received by editor(s): August 14, 2007
Posted: 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.
Copyright of article: Copyright 2009, American Mathematical Society

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