There is no degree invariant half-jump

Abstract

We prove that there is no degree invariant solution to Post's prob- lem that always gives an intermediate degree. In fact, assuming definable de- terminacy, if W is any definable operator on degrees such that a < a′ on a cone then W is low2 or high2 on a cone of degrees, i.e., there is a degree csuchthatW(a)′′ =a′′ foreverya≥corW(a)′′ =a′′′ foreverya≥c.