Definability in the Turing degrees
نویسندگان
چکیده
منابع مشابه
Definability in the Turing Degrees
Suppose that R is a countable relation on the Turing degrees. Then R can be defined in D, the Turing degrees with ≤T , by a first order formula with finitely many parameters. The parameters are built by means of a notion of forcing in which the conditions are essentially finite. The conditions in the forcing partial specify finite initial segments of the generic reals and impose a infinite cons...
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We prove that the ∆2 Turing degrees have a finite automorphism base. We apply this result to show that the automorphism group of DT (≤ 0′) is countable and that all its members have arithmetic presentations. We prove that every relation on DT (≤ 0′) induced by an arithmetically definable degree invariant relation is definable with finitely many ∆2 parameters and show that rigidity for DT (≤ 0′)...
متن کاملThe ∆2 Turing Degrees: Automorphisms and Definability
We prove that the ∆2 Turing degrees have a finite automorphism base. We apply this result to show that the automorphism group of DT (≤ 0′) is countable and that all its members have arithmetic presentations. We prove that every relation on DT (≤ 0′) induced by an arithmetically definable degree invariant relation is definable with finitely many ∆2 parameters and show that rigidity for DT (≤ 0′)...
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We prove that every countable relation on the enumeration degrees, E, is uniformly definable from parameters in E. Consequently, the first order theory of E is recursively isomorphic to the second order theory of arithmetic. By an effective version of coding lemma, we show that the first order theory of the enumeration degrees of the Σ2 sets is not decidable.
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We obtain the first nontrivial d.c.e. Turing approximation to the class of computably enumerable (c.e.) degrees. This depends on the following extension of the splitting theorem for the d.c.e. degrees: For any d.c.e. degree a, any c.e. degree b, if b < a, then there are d.c.e. degrees x0,x1 such that b < x0,x1 < a and a = x0 ∨ x1. The construction is unusual in that it is incompatible with uppe...
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ژورنال
عنوان ژورنال: Illinois Journal of Mathematics
سال: 1986
ISSN: 0019-2082
DOI: 10.1215/ijm/1256044641