Definability via Kalimullin pairs in the structure of the enumeration degrees

Abstract

We give an alternative definition of the enumeration jump operator. We prove that the class of total enumeration degrees and the class of low enumeration degrees are first order definable in the local substructure of the enumeration degree, consisting of the elements bounded by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold 0 prime Subscript e"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn mathvariant="bold">0</mml:mn> </mml:mrow> <mml:mi>e</mml:mi> </mml:msub> </mml:mrow> <mml:mo>′</mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">{\mathbf {0}_e}’</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .