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Maximal chains in the Turing degrees

Published online by Cambridge University Press:  12 March 2014

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

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
Copyright
Copyright © Association for Symbolic Logic 2007

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References

REFERENCES

[1]Abraham, Uri and Shore, Richard A., Initial segments of the degrees of size ℵ1, Israel journal of Mathematics, vol. 53 (1986), no. 1, pp. 151.CrossRefGoogle Scholar
[2]Barwise, Jon, Admissible Sets and Structures, Springer-Verlag, Berlin, 1975.CrossRefGoogle Scholar
[3]Boolos, George and Putnam, Hilary, Degrees of unsolvability of constructible sets of integers, this Journal, vol. 33 (1968), pp. 497513.Google Scholar
[4]Devlin, Keith J., Constructibility, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1984.CrossRefGoogle Scholar
[5]van Engelen, Fons, Miller, Arnold W., and Steel, John, Rigid Borel sets and better quasiorder theory, Logic and Combinatorics (Areata, California, 1985), Contemporary Mathematics, vol. 65, American Mathematical Society, Providence, RI, 1987, pp. 199222.CrossRefGoogle Scholar
[6]Jech, Thomas, Set Theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003.Google Scholar
[7]Jensen, R. B., The fine structure of the constructible hierarchy, Annals of Mathematical Logic, vol. 4 (1972), pp. 229308, erratum, ibid. 4 (1972), 443.CrossRefGoogle Scholar
[8]Lerman, Manuel, Degrees of Unsolvability, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1983, Local and global theory.CrossRefGoogle Scholar
[9]Martin, D. A., Borel determinacy. Annals of Mathematics (2), vol. 102 (1975), no. 2, pp. 363371.CrossRefGoogle Scholar
[10]Martin, D. A. and Solovay, R. M., Internal Cohen extensions, Annals of Mathematical Logic, vol. 2 (1970), no. 2, pp. 143178.CrossRefGoogle Scholar
[11]Miller, Arnold W., Infinite combinatorics and definability, Annals of Pure and Applied Logic, vol. 41 (1989), no. 2, pp. 179203.CrossRefGoogle Scholar
[12]Moschovakis, Yiannis N., Descriptive Set Theory, Studies in Logic and the Foundations of Mathematics, vol. 100, North-Holland Publishing, Amsterdam, 1980.Google Scholar
[13]Sacks, Gerald E., Degrees of Unsolvability, Princeton University Press, Princeton, N.J., 1963.Google Scholar
[14]Sacks, Gerald E., Higher Recursion Theory, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1990.CrossRefGoogle Scholar
[15]Simpson, Stephen G., Minimal covers and hyperdegrees, Transactions of the American Mathematical Society, vol. 209 (1975), pp. 4564.CrossRefGoogle Scholar
[16]Solovay, Robert M., On the cardinality of sets of reals, Foundations of Mathematics (Symposium commemorating Kurt Gödel, Columbus, Ohio, 1966), Springer, New York, 1969, pp. 5873.Google Scholar
[17]Solovay, Robert M., A model of set-theory in which every set of reals is Lebesgue measurable. Annals of Mathematics (2), vol. 92 (1970), pp. 156.CrossRefGoogle Scholar