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 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. ∗Research partially supported by the U.S. ARO through ACSyAM at the Mathematical Sciences Institute of Cornell University Contract DAAL03-91-C-0027, the IGC of Victoria University and the Marsden Fund for Basic Science under grant VIC-509. §Research partially supported by NSF Grant DMS-9503503 and the U.S. ARO through ACSyAM at the Mathematical Sciences Institute of Cornell University Contract DAAL0391-C-0027.