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Definable incompleteness and Friedberg splittings

  • Russell Miller (a1)
Abstract

We define a property R(A0, A1) in the partial order of computably enumerable sets under inclusion, and prove that R implies that A0 is noncomputable and incomplete. Moreover, the property is nonvacuous. and the A0 and A1 which we build satisfying R form a Friedberg splitting of their union A, with A1 prompt and A promptly simple. We conclude that A0 and A1 lie in distinct orbits under automorphisms of , yielding a strong answer to a question previously explored by Downey, Stob, and Soare about whether halves of Friedberg splittings must lie in the same orbit.

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Corresponding author
Department of Mathematics, Cornell University, Ithaca, New York 14853, USA, E-mail: russell@math.cornell.edu
References
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[1]Cholak, P., Automorphisms of the lattice of recursively enumerable sets, Memoirs of the American Mathematical Society, vol. 113 (1995), no. 541.
[2]Cholak, P., Downey, R., and Stob, M., Automorphisms of the lattice of recursively enumerable sets: Promptly simple sets, Transactions of the American Mathematical Society, vol. 332 (1993). pp. 555569.
[3]Downey, R. and Stob, M., Jumps of hemimaximal sets, Zeitschrift für Math. Logik Grundlagen, vol. 37 (1991), pp. 113120.
[4]Downey, R. and Stob, M., Automorphisms of the lattice of recursively enumerable sets: Orbits, Advances in Mathematics, vol. 92 (1992), pp. 237265.
[5]Downey, R. and Stob, M., Friedberg splittings of recursively enumerable sets, Annals of Pure and Applied Logic, vol. 59 (1993), pp. 175199.
[6]Friedberg, R. M., Two recursively enumerable sets of incomparable degrees of unsolvability, Proceedings of the National Academy of Sciences, USA, vol. 43 (1957), pp. 236238.
[7]Harrington, L. and Soare, R. I., Post's program and incomplete recursively enumerable sets, Proceedings of the National Academy of Sciences, USA, vol. 88 (1991), pp. 1024210246.
[8]Harrington, L. and Soare, R. I., The Δ30-automorphism method and noninvariant classes of degrees, Journal of the American Mathematical Society, vol. 9 (1996). pp. 617666.
[9]Harrington, L. and Soare, R. I., Definable properties of the computably enumerable sets, Annals of Pure and Applied Logic, vol. 94 (1998), pp. 97125.
[10]Jockusch, C. G. Jr., Review of Lerman [11], Mathematical Reviews, vol. 45 (1973). #3200.
[11]Lerman, M., Some theorems on r-maximal sets and major subsets of recursively enumerable sets, this Journal, vol. 36 (1971). pp. 193215.
[12]Maass, W. and Stob, M., The intervals of the lattice of recursively enumerable sets determined by major subsets, Annals of Pure and Applied Logic, vol. 24 (1983), pp. 189212.
[13]Muchnik, A. A., On the unsolvability of the problem of reducibilily in the theory of algorithms, Dokl. Akad. Nauk SSSR, N. S., vol. 109 (1956). pp. 194197, in Russian.
[14]Myhill, J., The lattice of recursively enumerable sets, this Journal, vol. 21 (1956). pp. 215, 220.
[15]Rogers, H. Jr., Theory of recursive functions and effective computability, The MIT Press, Cambridge, Massachusetts, 1987.
[16]Soare, R. I., Recursively enumerable sets and degrees, Springer-Verlag, New York, 1987.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
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