Independence results on the global structure of the Turing degrees
Authors:
Marcia J. Groszek and Theodore A. Slaman
Journal:
Trans. Amer. Math. Soc. 277 (1983), 579588
MSC:
Primary 03D30; Secondary 03E35
MathSciNet review:
694377
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Abstract 
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Abstract: From CON(ZFC) we obtain: 1. CONZFC is arbitrarily large there is a locally finite upper semilattice of size which cannot be embedded into the Turing degrees as an upper semilattice). 2. CONZFC is arbitrarily large there is a maximal independent set of Turing degrees of size ).
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 J. Baumgartner and R. Laver, Iterated perfect set forcing, Ann. Math. Logic 17 (1979), 271288. MR 556894 (81a:03050)
 [2]
 S. C. Kleene and E. L. Post, The upper semilattice of degrees of recursive unsolvability, Ann. of Math. (2) 59 (1954), 379407. MR 0061078 (15:772a)
 [3]
 P. Cohen, Set theory and the continuum hypothesis, Benjamin, New York, 1966. MR 0232676 (38:999)
 [4]
 A. H. Lachlan, Distributive initial segments of the degrees of unsolvability, Math. Logik Grundlag. Math. 14 (1968), 457472. MR 0237331 (38:5620)
 [5]
 A. H. Lachlan and R. Lebeuf, Countable initial segments of the degrees of unsolvability, J. Symbolic Logic 41 (1976), 289300. MR 0403937 (53:7746)
 [6]
 M. Lerman, Initial segments of the degrees of unsolvability, Ann. of Math. (2) 93 (1971), 365389. MR 0307893 (46:7008)
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 J. M. Rubin, The existence of an initial segment of the Turing degrees, Notices Amer. Math. Soc. 26 (1979), Abstract #79TA168, p. A425.
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 , Distributive uncountable initial segments of the degrees of unsolvability, Notices Amer. Math. Soc. 26 (1979), Abstract #79TE74, p. A619.
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 G. E. Sacks, Degrees of unsolvability, Ann. of Math. Studies, no. 55, Princeton Univ. Press, Princeton, N.J., 1963. MR 0186554 (32:4013)
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 , Forcing with perfect closed sets, Axiomatic Set Theory (Dana Scott, editor), Proc. Sympos. Pure Math., vol. 13, Amer. Math. Soc., Providence, R.I., 1971, pp. 331355. MR 0276079 (43:1827)
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 S. G. Simpson, Degrees of unsolvability: a survey of results, Handbook of Mathematical Logic (J. Barwise, editor), NorthHolland, Amsterdam, 1977.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198306943774
PII:
S 00029947(1983)06943774
Keywords:
Turing degrees,
forcing,
uncountable embeddings,
independent sets
Article copyright:
© Copyright 1983
American Mathematical Society
