It's well known that Reinhardt cardinals are inconsistent, provided that we have access to axiom of choice, but, as far as I know, we are clueless about this when we don't assume choice. For me, the fact that Reihardt cardinals are inconsistent with choice already makes them, in a way, "wrong", that is, I'd believe personally that Reinhardt cardinals cannot exist in $V$ anyways. However, I have never seen any claims like this. So my question is:



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> Is it believed that Reinhardt cardinals can be consistent without choice? Are there any solid arguments for/against them? (other than "we haven't found them inconsistent, so they should be consistent")
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Thanks in advance.


Note: I added soft-question tag because this question is more about philosophical views. Feel free to remove it if you think otherwise.



What are the current views on consistency of Reinhardt cardinals without AC?
The following theorems of Woodin may be related:


**Theorem.** ($ZF$) Assume that $ZFC$ proves the $HOD$ Conjecture. Suppose $\delta$ is an extendible cardinal. Then for all $ \lambda>\delta$ there is no non-trivial
elementary embedding $j : V\_{\lambda+2}\to V\_{\lambda+2}$ 


Thus (assuming that $ZFC$ proves the $HOD$ Conjecture) one nearly has a
proof of Kunen's inconsistency theorem without using the Axiom of Choice.


**Theorem.** Assume $ZF$+ there exists a Reinhardt cardinal + there exists a proper class of supercompact cardinals is consistent. Then there exists a genric extension of the universe which satisfies $ZF$ + the axiom of choice + there exists a proper class of supercompact cardinals, and such that in it Woodin's $HOD$ conjecture fails.


As far as I know, Woodin believes the $HOD$ conjecture is true (at least the current methods can not be used to solve the problem), so by the above theorem we may expect to show that at least **"$ZF$+ there exists a Reinhardt cardinal + there exists a proper class of supercompact cardinals"** is not consistent. 


