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The theory of the recursively enumerable weak truth-table degrees is undecidable

  • Klaus Ambos-Spies (a1), André Nies (a1) and Richard A. Shore (a2)

Abstract

We show that the partial order of -sets under inclusion is elementarily definable with parameters in the semilattice of r.e. wtt-degrees. Using a result of E. Herrmann, we can deduce that this semilattice has an undecidable theory, thereby solving an open problem of P. Odifreddi.

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[ASp,N?]Ambos-Spies, K. and Nies, A., The theory of the polynomial many-one degrees of recursive sets is undecidable (in preparation).
[ASp.Sh?]Ambos-Spies, K. and Shore, R., Undecidability and 1-types in the r.e. degrees (in preparation).
[ASp,So89]Ambos-Spies, K. and Soare, R. I., The recursively enumerable degrees have infinitely many one-types, Annals of Pure and Applied Logic, vol. 44, pp. 123.
[Bu,McK81]Burris, S. and McKenzie, R., Decidability and Boolean representation, Memoir no. 246, American Mathematical Society, Providence, Rhode Island.
[Ha,Sh82]Harrington, L. and Shelah, S., The undecidability of the recursively enumerable degrees, Bulletin (New Series) of the American Mathematical Society, vol. 6, pp. 7980 (research announcement).
[He83]Herrmann, E., Definable Boolean pairs and the lattice of recursively enumerable sets, Proceedings of the first Easter conference on model theory, Seminarberichte no. 49, Sektion Mathematik, Humboldt-Universität, Berlin, pp. 4267.
[He84]Herrmann, E., The undecidability of the elementary theory of the lattice of recursively enumerable sets, Proceedings of the second Frege conference at Schwerin, GDR, 1984 (Wechsung, G., editor), Mathematische Forschung, Band 20, Akademie-Verlag, Berlin, pp. 6672.
[Ht,S90]Haught, C. A. and Shore, R. A., Undecidability and initial segments of the (r.e.) tt-degrees, this Journal, vol. 55, pp. 9871006.
[La72]Lachlan, A. H., Embedding nondistributive lattices in the recursively enumerable degrees, Conference in mathematical logic—London '70, Lecture Notes in Mathematics, vol. 255, Springer-Verlag, Berlin, pp. 149177.
[Ld,Sa75]Ladner, R. and Sasso, S., The weak truth-table degrees of r.e. sets, Annals of Mathematical Logic, vol. 8, pp. 429448.
[Od81]Odifreddi, P., Strong reducibilities, Bulletin (New Series) of the American Mathematical Society, vol. 4, pp. 3786.
[Od89]Odifreddi, P., Classical recursion theory, North-Holland, Amsterdam.
[So87]Soare, R. I., Recursively enumerable sets and degrees, Springer-Verlag, Berlin.

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