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On the Σ2-theory of the upper semilattice of Turing degrees

  • Carl G. Jockusch (a1) and Theodore A. Slaman (a2)

A first-order sentence Φ is Σ2 if there is a quantifier-free formula Θ such that Φ has the form . The Σ2-theory of a structure for a language ℒ is the set of Σ2-sentences of true in . It was shown independently by Lerman and Shore (see [Le, Theorem VII.4.4]) that the Σ2-theory of the structure = 〈D, ≤ 〉 is decidable, where D is the set of degrees of unsolvability and ≤ is the standard ordering of D. This result is optimal in the sense that the Σ3-theory of is undecidable, a result due to J. Schmerl. (For a proof, see [Le, Theorem VII.4.5]. As Lerman has pointed out, this proof should be corrected by defining θσ to be ∀1(x) rather than ∀x(ψ(x)σ1(x)).) Nonetheless, in this paper we extend the decidability result of Lerman and Shore by showing that the Σ2-theory of is decidable, where ⋃ is the least upper bound operator and 0 is the least degree. Of course ⋃ is definable in , but many interesting degree-theoretic results are expressible as Σ2-sentences in the language of but not as Σ2-sentences in the language of . For instance, Simpson observed that the Posner-Robinson cupping theorem could be used to show that for any nonzero degrees a, b, there is a degree g such that bag, and bg (see [PR, Corollary 6]). However, the Posner-Robinson technique does not seem to suffice to decide the Σ2-theory of . We introduce instead a new method for coding a set into the join of two other sets and use it to decide this theory.

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[La]Lachlan, A. H., The elementary theory of recursively enumerable sets, Duke Mathematical Journal, vol. 35 (1968), pp. 123146.
[Le]Lerman, M., Degrees of unsolvability, Perspectives in Mathematical Logic, Omega Series, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1983.
[LS]Lerman, M. and Soare, R. I., A decidable fragment of the elementary theory of the lattice of recursively enumerable sets, Transactions of the American Mathematical Society, vol. 257(1980), pp. 137.
[PR]Posner, D. and Robinson, R., Degrees joining to 0′, this Journal, vol. 46 (1981), pp. 705713.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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