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Enumeration reducibility was introduced by Friedberg and Rogers in 1959 as a positive reducibility between sets. The enumeration degrees provide a wider context in which to view the Turing degrees by allowing us to use any set as an oracle instead of just total functions. However, in spite of the fact that there are several applications of enumeration… (More)
We show that the first order theory of the Σ 0 2 s-degrees is undecidable. Via isomorphism of the s-degrees with the Q-degrees, this also shows that the first order theory of the Π 0 2 Q-degrees is undecidable. Together with a result of Nies, the proof of the undecidability of the Σ 0 2 s-degrees yields a new proof of the known fact (due to Downey, LaForte… (More)
For any enumeration degree a let D s a be the set of s-degrees contained in a. We answer an open question of Watson by showing that if a is a nontrivial Σ 0 2-enumeration degree, then D s a has no least element. We also show that every countable partial order embeds into D s a .
For any enumeration degree a let D s a be the set of s-degrees contained in a. We answer an open question of Watson by showing that if a is a nontrivial Σ 0 2-enumeration degree, then D s a has no least element. We also show that every countable partial order embeds into D s a. Finally, we construct Σ 0 2-sets A and B such that B ≤ e A but for every X ≡ e… (More)
Tunnel junctions could have a great impact on gallium nitride and aluminum nitride-based devices such as light-emitting diodes and lasers by overcoming critical challenges related to hole injection and p-contacts. This paper demonstrates the use of GdN nanoislands to enhance interband tunneling and hole injection into GaN p-n junctions by several orders of… (More)
When working with a mathematical structure, it is natural to ask how complicated the first order theory of the structure is. In the case of structures that can be interpreted in first order arithmetic, this is equivalent to asking if the theory is as complex as possible, namely is it as complex as first order arithmetic. Restricting our attention to the… (More)