# A Hot Betting On HOD

Remark: This question is based on an open question at the end of a paper by Hamkins, Kirmayer, and Perlmutter: "Generalizations of the Kunen Inconistency".

$HOD$ as an inner model of $ZFC$ lies between $L$ and $V$. But its nature shows a wide degree of flexibility between $L$ -like and $V$-like behaviors up to different problems.

One of the problems about any inner model of $ZFC$ is consistency of existence of a non-trivial self elementary embedding. In this direction the following facts are well known:

(a) There is no non-trivial elementary embedding from $V$ to $V$.

(b) Existence of a non-trivial elementary embedding from $L$ to $L$ is an acceptable large cardinal axiom.

Here the open question is:

Question (1): Is the existence of a non-trivial elementary embedding from $HOD$ to $HOD$ an acceptable large cardinal axiom?

I usually discuss with my colleagues about it. I believe that $HOD$ in this problem has a $L$ - like behavior and existence of a non-trivial elementary embedding from $HOD$ to $HOD$ is an acceptable (not too) large cardinal axiom but one of my friends insists that the facts shows there is no such embedding. Finally we bet on it. Here I want to know about any known partial result around the open problem. So:

Question (2): Please introduce any known fact related to the possible answer of question (1) and explain that which one of the scenarios (a) or (b) for $HOD$ seems more possible by this fact.

• If I understand correctly, then one can get a model of ZFC with an elementary embedding from HOD into HOD assuming that a strong large cardinal axiom related to Reinhardt cardinals is consistent with ZF (see J1 in en.wikipedia.org/wiki/Reinhardt_cardinal). However, currently (I believe) the tide of general opinion is against Reinhardts in ZF; so this might not answer your question. – Noah Schweber Oct 26 '13 at 21:55
• Related: the consistency with ZFC of "HOD into HOD" is currently unknown: cantorsattic.info/Kunen_inconsistency. – Noah Schweber Oct 26 '13 at 21:57
• @Noah: Thanks. What is your personal idea about it? Is it consistent or inconsistent? – user36136 Oct 26 '13 at 22:02
• Woodin conjectures that the answer is no, there is no non-trivial elementary embedding from HOD to itself. This follows from the existence of an extendible cardinal, assuming a separate conjecture known as the HOD conjecture. There is a set of notes titled the HOD Dichotomy that goes into these matters in some detail and the notes are actually pretty accessible. You may also want to look at a paper by Hamkins, Kirmayer, and Perlmutter: "Generalizations of the Kunen Inconistency" – Everett Piper Oct 26 '13 at 23:01
• @Everett: So existence of such embedding refutes the existence of some very large cardinals like extendibles. Now it seems I should change my side in the betting! – user36136 Oct 26 '13 at 23:06

• Implicit in your belief is somehow the notion that $\sf HOD$ should be very close to $V$, even if choice fails in $V$. Going to the extreme, consider Gitik's model where all the limit ordinals have countable cofinality. How could that model be anywhere close to $\sf HOD$? (While that model certainly doesn't serve as an example to an elementary embedding $j\colon V\to V$, it is still an example of how very far $V$ and $\sf HOD$ can be.) – Asaf Karagila Oct 28 '13 at 21:14
• @Ali: I'm still trying to work my intuition on that, but if you'd put a gun to my whisky collection and force me to make a choice, I'd save my Glen Livet and choose that $V$ can be very far from $\sf HOD$. – Asaf Karagila Oct 28 '13 at 23:35