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Transactions of the American Mathematical Society

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Generic sets and minimal $ \alpha $-degrees


Author: C. T. Chong
Journal: Trans. Amer. Math. Soc. 254 (1979), 157-169
MSC: Primary 03D60; Secondary 03D30
DOI: https://doi.org/10.1090/S0002-9947-1979-0539912-0
MathSciNet review: 539912
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Abstract: A non-$ \alpha $-recursive subset G of an admissible ordinal $ \alpha $ is of minimal $ \alpha $-degree if every set of strictly lower $ \alpha $-degree than that of G is $ \alpha $-recursive. We give a characterization of regular sets of minimal $ \alpha $-degree below $ 0'$ via the notion of genericity. We then apply this to outline some 'minimum requirements' to be satisfied by any construction of a set of minimal $ \aleph _\omega ^L$-degree below $ 0'$.


References [Enhancements On Off] (What's this?)

  • [1] C. T. Chong, An $ \alpha $-finite injury method of the unbounded type, J. Symbolic Logic 41 (1976), 1-17. MR 0476456 (57:16019)
  • [2] S. B. Cooper, Minimal degrees and the jump operator, J. Symbolic Logic 38 (1973), 249-271. MR 0347572 (50:75)
  • [3] W. Maass, On minim pairs and minimal degrees in higher recursion theory, Arch. Math. Logik Grundlagenforsch. 18 (1976), 169-186. MR 0485289 (58:5136)
  • [4] G. E. Sacks, Forcing with perfect closed sets, Proc. Sympos. Pure Math., vol. 13, Amer. Math. Soc., Providence, R. I., 1971. pp. 331-355. MR 0276079 (43:1827)
  • [5] G. E. Sacks and S. G Simpson, The $ \alpha $-finite injury method, Ann. Math. Logic 4 (1972), 343-367. MR 0369041 (51:5277)
  • [6] R. A. Shore, Minimal $ \alpha $-degrees, Ann. Math. Logic 4 (1972), 393-414. MR 0369042 (51:5278)
  • [7] -, Splitting an $ \alpha $-recursively enumerable set, Trans. Amer. Math. Soc. 204 (1975), 65-78. MR 0379154 (52:60)
  • [8] C. Spector, On degrees of recursive unsolvability, Ann. of Math. (2) 64 (1956), 581-592. MR 0082457 (18:552d)
  • [9] C. E. M. Yates, Initial segments of the degrees of unsolvability, Mathematical Logic and Foundations of Set Theory (Y. Bar-Hillel, Editor), North-Holland, Amsterdam, 1970, pp. 63-83. MR 0269505 (42:4400)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1979-0539912-0
Keywords: Admissible ordinal, minimal $ \alpha $-degree, generic set
Article copyright: © Copyright 1979 American Mathematical Society

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