From Mr. Energy:
Why? I think I made my position clear enough: I stated a consistent Maximality Criterion and based on my proof (with co-authors) of its consistency I have the impression that this Criterion contradicts supercompacts (not just extendibles). So that leads to a tentative rejection of supercompacts until the situation changes through further understanding of further Maximality Criteria. It’s analagous to what happened with the IMH: It led to a tentative rejection of inaccessibles, but then when Vertical Maximality was taken into account, it became obvious that the IMH# was a better criterion than the IMH and the IMH# is compatible with inaccessibles and more.
I also think that the Maximality Criterion I stated could be made much stronger, which I think is only possible if one denies the existence of supercompacts. (Just a conjecture, no theorem yet.)
Looks like I have been nominated long ago (smile) to try to turn this controversy into something readily digestible – and interesting – for everybody.
A main motivator for me in this arguably unproductive traffic is to underscore the great value of real time interaction. Bad ideas can be outed in real time! Bad ideas can be reformulated as reasonable ideas in real time!! Good new ideas can emerged in real time!!! What more can you want? Back to this situation.
This thread is now showing even more clearly the pitfalls of using unanalyzed flowery language like “Maximality Criterion” to try to draw striking conclusions (technical advances not yet achieved, but perhaps expected). Nobody would bother to complain if the striking conclusions were compatible with existing well accepted orthodoxy.
So what is really being said here is something like this:
“My (Mr. Energy) fundamental thinking about the set theoretic universe is so wise that under anticipated technical advances, it is sufficient to overthrow long established and generally accepted orthodoxy”.
What is so unusual here is that this unwarranted arrogance is so prominently displayed in a highly public environment with several of the most well known scholars in relevant areas actively engaged!
What was life like before email? We see highly problematic ideas being unravelled in real time.
What would a rational person be putting forward? Instead of the arrogant
*Maximality Criteria tells us that HOD is much smaller than V and this (is probably going to be shown in the realistic future to) refutes certain large cardinal hypotheses*
the entirely reasonable
**Certain large cardinal hypotheses (are probably going to be shown in the realistic future to) imply that HOD has similarities to V. Such similarities cannot be proved or refuted in ZFC. This refutes certain kinds of formulations of “Maximality in higher set theory, under relevant large cardinal hypotheses.**
and then remark something like this:
***The notion “intrinsic maximality of the set theoretic universe” is in great need of clear elucidation. Many formulations lead to inconsistencies or refutations of certain large cardinal hypotheses. We hope to find a philosophically coherent analysis of it from first principles that may serve as a guide to the appropriateness of many set theoretic hypotheses. In particular, the use of HOD in formulations can be criticized, and raises a number of unresolved issues.***
Again, what was life like before email? We might have been seeing students and postdocs running around Europe opening claiming to refute various large cardinal hypotheses!