INITIAL SEGMENTS OF THE ENUMERATION DEGREES

Abstract Using properties of ${\cal K}$ -pairs of sets, we show that every nonzero enumeration degree a bounds a nontrivial initial segment of enumeration degrees whose nonzero elements have all the same jump as a . Some consequences of this fact are derived, that hold in the local structure of the enumeration degrees, including: There is an initial segment of enumeration degrees, whose nonzero elements are all high; there is a nonsplitting high enumeration degree; every noncappable enumeration degree is high; every nonzero low enumeration degree can be capped by degrees of any possible local jump (i.e., any jump that can be realized by enumeration degrees of the local structure); every enumeration degree that bounds a nonzero element of strictly smaller jump, is bounding; every low enumeration degree below a non low enumeration degree a can be capped below a .