Online research in philosophy

In [9]. Yates proved the existence of a Turing degree a such that 0. 0′ are the only c.e. degrees comparable with it. By Slaman and Steel [7], every degree below 0′ has a 1-generic complement, and as a consequence. Yates degrees can be 1-generic, and hence can be low. In this paper, we prove that Yates degrees occur in every jump class.

In this paper, we study the bi-isolation phenomena in the d.c.e. degrees and prove that there are c.e. degrees c₁ < c₂ and a d.c.e. degree d ∈ (c₁, c₂) such that (c₁, d) and (d, c₂) contain no c.e. degrees. Thus, the c.e. degrees between c₁ and c₂ are all incomparable with d. We also show that there are d.c.e. degrees d₁ < d₂ such that (d₁, d₂) contains a unique c.e. degree.

Say that (a, d) is an isolation pair if a is a c.e. degree, d is a d.c.e. degree, a < d and a bounds all c.e. degrees below d. We prove that there are an isolation pair (a, d) and a c.e. degree c such that c is incomparable with a, d, and c cups d to o', caps a to o. Thus, {o, c, d, o'} is a diamond embedding, which was first proved by Downey in [9]. Furthermore, (...) combined with Harrington-Soare continuity of capping degrees, our result gives an alternative proof of N5 embedding. (shrink)