Independence results on the global structure of the Turing degrees
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- by Marcia J. Groszek and Theodore A. Slaman PDF
- Trans. Amer. Math. Soc. 277 (1983), 579-588 Request permission
Abstract:
From $\text {CON(ZFC)}$ we obtain: 1. $\text {CON}(\text {ZFC} + 2^\omega$ is arbitrarily large $+$ there is a locally finite upper semilattice of size ${\omega _2}$ which cannot be embedded into the Turing degrees as an upper semilattice). 2. $\text {CON}(\text {ZFC} + 2^\omega$ is arbitrarily large $+$ there is a maximal independent set of Turing degrees of size ${\omega _1}$).References
- James E. Baumgartner and Richard Laver, Iterated perfect-set forcing, Ann. Math. Logic 17 (1979), no. 3, 271–288. MR 556894, DOI 10.1016/0003-4843(79)90010-X
- S. C. Kleene and Emil L. Post, The upper semi-lattice of degrees of recursive unsolvability, Ann. of Math. (2) 59 (1954), 379–407. MR 61078, DOI 10.2307/1969708
- Paul J. Cohen, Set theory and the continuum hypothesis, W. A. Benjamin, Inc., New York-Amsterdam, 1966. MR 0232676
- A. H. Lachlan, Distributive initial segments of the degrees of unsolvability, Z. Math. Logik Grundlagen Math. 14 (1968), 457–472. MR 237331, DOI 10.1002/malq.19680143002
- A. H. Lachlan and R. Lebeuf, Countable initial segments of the degrees of unsolvability, J. Symbolic Logic 41 (1976), no. 2, 289–300. MR 403937, DOI 10.2307/2272227
- Manuel Lerman, Initial segments of the degrees of unsolvability, Ann. of Math. (2) 93 (1971), 365–389. MR 307893, DOI 10.2307/1970779 J. M. Rubin, The existence of an ${\omega _1}$ initial segment of the Turing degrees, Notices Amer. Math. Soc. 26 (1979), Abstract #79T-A168, p. A-425. —, Distributive uncountable initial segments of the degrees of unsolvability, Notices Amer. Math. Soc. 26 (1979), Abstract #79T-E74, p. A-619.
- Gerald E. Sacks, Degrees of unsolvability, Princeton University Press, Princeton, N.J., 1963. MR 0186554
- Gerald E. Sacks, Forcing with perfect closed sets, Axiomatic Set Theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967) Amer. Math. Soc., Providence, R.I., 1971, pp. 331–355. MR 0276079 S. G. Simpson, Degrees of unsolvability: a survey of results, Handbook of Mathematical Logic (J. Barwise, editor), North-Holland, Amsterdam, 1977.
Additional Information
- © Copyright 1983 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 277 (1983), 579-588
- MSC: Primary 03D30; Secondary 03E35
- DOI: https://doi.org/10.1090/S0002-9947-1983-0694377-4
- MathSciNet review: 694377