Consider the following theorem of Koepke-Koerwien-Siders:
"A set x of ordinals is ordinal computable [either by ordinal Turing machines or ordinal register machines--my comment] if and only if it is an element of the constructible universe L"
Consider also the following theorem from Kunen's book SET THEORY: An Introduction to Independence Proofs:
Theorem 3.5 (Kunen, 1980, pg 171): If M is any transitive proper class model of ZF-P [P is the power set axiom--my comment], L=L^(M) is a proper subset of M.
Kunen uses this theorem to prove (assuming M=HOD) that L is a proper subset of HOD.
Assuming V=HOD, are there sets of ordinals definable in HOD that are not ORM (Ordinal Register Machine--respectively Ordinal Turing Machine) computable? I ask the question because HOD as does OD, relies on definability from a finite set of ordinal parameters. It should be noted that in problem 22 of Kunen (1980) Chapter 6, one is asked to prove that for L^(n) (that is, L defined in n'th-order logic, n>=2) that L^(n)=HOD). Assuming that the problem is not ill-formed, it can be deemed a theorem. This theorem might be deemed a mitigating factor.