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Inspired by this question: What are the known implications of "There exists a Reinhardt cardinal" in the theory "ZF + j"?

Definitions:

$\delta$ is Berkeley iff for every $\alpha\lt\delta$ and transitive set $M$ such that $\delta\in M$, there is some $j: M\prec M$ such that $\alpha\lt\text{crit}j\lt\delta$.

$\delta$ is club Berkeley iff for every club $C\subseteq\delta$ and transitive set $M$ such that $\delta\in M$, there is some $j: M\prec M$ such that $\text{crit}j\in C$.

$\delta$ is limit club Berkeley if $\delta$ is club Berkeley and a limit of Berkeley cardinals.

Question 1: What is the progress on anaylizing Berkeley cardinals?

Have any of the major open problems since the introduction of Berkeley cardinals been solved? Are club Berkeley cardinals larger than Berkeley cardinals? What is the effect of increasing levels of Choice on Berkeley cardinals?

Berkeley cardinals are infamously large large cardinals. The original paper Large Cardinals Beyond Choice where they were introduced, left open 7 problems, 4 of which were related to Berkeley cardinals. Have any of these been solved? Have any new techniques or results created hope that they might be solved in the foreseeable future?

Question 2: Can we go bigger?

Are there any large cardinals that imply the consistency of (Limit club) Berkeley cardinals? If they exist, are there any interesting results that these larger cardinals imply.

This question is simple. How big can we go? For this particular question, I am willing to accept more "off-the-cuff" answers. Large cardinals developed in private communications, or even developed specifically to answer this question. All the criteria are is that they have to be stronger than club Berkeley cardinals, they can't be trivial extensions, they can't be obvious, and they have to either have clear reasoning behind their development, or at least you need to be able to do something interesting with them.

Question 3: Can we rescue Berkeley cardinals?

Are there any variants of Berkeley cardinals that can survive even in the context of Woodin's Weak $HOD$ conjecture?

Consider the nightmare scenario: Over the next decade or two, more and more results related to Reinhardt and Berkeley cardinals are proved, the open problems are resolved, and an interesting structural theory is developed. Than the weak $HOD$ conjecture is proved and that entire hierarchy is wiped out.

It is really this nightmare scenario that is the motivation for my question. Is there any weakening of Berkeley cardinals that would allow us to save any structure theory developed, in much the same way as set theory was rescued after Russel's paradox? The obvious answer might be the $HOD$-variants, but I believe even these can be wiped out. Assuming that is not the case, the question is do these really have all the properties that Berkeley cardinals have.

For example:

Are $HOD$-Super Reinhardt cardinals stronger than $HOD$-Reinhardt cardinals?

Better yet, perhaps some of the open problems that cannot be solved with classical Berkeley cardinals, can be resolved with these new ones!

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    $\begingroup$ Would the downvoter like to explain the problem they had with the question? $\endgroup$
    – Master
    Dec 25, 2020 at 3:00
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    $\begingroup$ I didn't downvote, but if I had to guess, you're asking 6 questions. I suppose some people find that off-putting. (You know, like someone who stands up after a talk and rattles off a series of questions.) $\endgroup$
    – Andrej Bauer
    Dec 25, 2020 at 10:32
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    $\begingroup$ I should add to what @Andrej wrote that these questions are kind of loaded as well. Nobody interacted with large cardinals beyond choice for a long time (in part, I suspect, since for over two decades Woodin hinted that his refutation is almost ready), and so a lot of these questions remain open and unstudied. Asking open questions on MathOverflow always felt to me as an exercise in futility. $\endgroup$
    – Asaf Karagila ♦
    Dec 25, 2020 at 16:19
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    $\begingroup$ Other than "Large cardinals beyond choice", there are Cutolo's papers "Berkeley cardinals and the structure of $L(V_{\delta+1})$" (JSL) (is that what you're referring to in the comment above?) and "The cofinality of the least Berkeley cardinal and the extent of dependent choice" (MLQ). There's also "Choiceless large cardinals and set-theoretic potentialism" by Cutolo and Hamkins (preprint on arXiv). $\endgroup$
    – Farmer S
    Dec 26, 2020 at 21:04
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    $\begingroup$ And more generally, $N$-Berkeley cardinals appear in Cutolo's paper "$N$-Berkeley cardinals and weak extender models" (to appear in JSL). There's also an old fact on "rank-Berkeley" cardinals mentioned in the last section of Goldberg's and my paper "Periodicity in the cumulative hierarchy" (preprint on arXiv). $\endgroup$
    – Farmer S
    Dec 26, 2020 at 21:05

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