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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.

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    $\begingroup$ Kunen does not use $\subset$ to mean "proper subset". $\endgroup$
    – François G. Dorais
    Commented Jan 20, 2013 at 14:18
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    $\begingroup$ I'm not sure I understand what the question is, but $0^\#$, if it exists, is a hereditarily ordinal-definable set of ordinals (indeed, a $\Pi_1$-definable set of integers) that isn't constructible (hence not ordinal machine computable). $\endgroup$
    – Gro-Tsen
    Commented Jan 20, 2013 at 15:05
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    $\begingroup$ Theorem 3.5 as quoted is wrong, but Francois's comment explains at least part of what's wrong with it. L is a subclass of any such M, but not necessarily a proper subclass. In particular, M could be L. $\endgroup$
    – Andreas Blass
    Commented Jan 20, 2013 at 18:44
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    $\begingroup$ To amplify Gro-Tsen's comment: If $0^{\#}$ exists then $L[0^{\#}]$ satisfies V=HOD and $0^{\#}$ is definable in it but not constructible. Finally, concerning the last sentence in the question: What is being mitigated? I see nothing in need of mitigation except the mistranslation of Kunen's $\subset$ as "proper subset". $\endgroup$
    – Andreas Blass
    Commented Jan 20, 2013 at 18:49
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    $\begingroup$ Thomas, all of these questions have well-known answers. I think you just need to do a little more careful research. In addition to Kunen, check out what Jech (Set Theory) and Kanamori (Higher Infinite) say on L, HOD, and other inner models. $\endgroup$
    – François G. Dorais
    Commented Jan 21, 2013 at 18:41

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