The high/low hierarchy in the local structure of the ω-enumeration degrees
نویسندگان
چکیده
This paper gives two definability results in the local theory of the ω-enumeration degrees. First we prove that the local structure of the enumeration degrees is first order definable as a substructure of the ω-enumeration degrees. Our second result is the definability of the the classes Hn and Ln of the highn and lown ω-enumeration degrees. This allows us to deduce that the first order theory of true arithmetic is interpretable in the local theory of the ω-enumeration degrees.
منابع مشابه
Splitting and nonsplitting in the Σ20 enumeration degrees
This paper continues the project, initiated in [ACK], of describing general conditions under which relative splittings are derivable in the local structure of the enumeration degrees, for which the Ershov hierarchy provides an informative setting. The main results below include a proof that any high total e-degree below 0′ e is splittable over any low e-degree below it, a non-cupping result in ...
متن کاملOn the jump classes of noncuppable enumeration degrees
We prove that for every Σ2 enumeration degree b there exists a noncuppable Σ2 degree a > 0e such that b ′ ≤e a′ and a′′≤e b′′. This allows us to deduce, from results on the high/low jump hierarchy in the local Turing degrees and the jump preserving properties of the standard embedding ι : DT → De, that there exist Σ2 noncuppable enumeration degrees at every possible—i.e. above low1—level of the...
متن کاملThe Jump Operator on the Ω-enumeration Degrees
The jump operator on the ω-enumeration degrees is introduced in [11]. In the present paper we prove a jump inversion theorem which allows us to show that the enumeration degrees are first order definable in the structureDω ′ of the ω-enumeration degrees augmented by the jump operator. Further on we show that the groups of the automorphisms of Dω ′ and of the enumeration degrees are isomorphic. ...
متن کاملThe Jump Hierarchy in the Enumeration Degrees
We show that all levels of the jump hierarchy are first order definable in the local structure of the enumeration degrees.
متن کاملNon-cuppable enumeration degrees via finite injury
We exhibit finite injury constructions of a high Σ2 enumeration degree incomparable with all intermediate ∆2 enumeration degrees, as also of both an upwards properly Σ2 high and a low2 noncuppable Σ 0 2 enumeration degree†. We also outline how to apply the same methods to prove that, for every Σ2 enumeration degree b there exists a noncuppable degree a such that b′ ≤ a′ and a′′ ≤ b′′, thus show...
متن کامل