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 low2 or high2 on a cone of degrees, i.e., there is a degree c such that W (a)′′ = a′′ for every a ≥ c or W (a)′′ = a′′′ for every a ≥ c. A striking phenomenon...