THERE IS NO DEGREE INVARIANT HALF-JUMP

We prove that there is no degree invariant solution to Post's problem that always gives an intermediate degree. In fact, assuming definable determinacy, if W is any definable operator on degrees such that a < W (a) < a' on a cone then W is low(2) or high(2) on a cone of degrees, i .e., there is a degree c such that W(a)'' = a '' for every a greater t han or equal to c or W(a)'' = a''' for every greater than or equal to c.