There’s quite a bit of terminology being used here that most people, including me, are not familiar with.
Parasitic (simple generally understandable examples, please), extendible cardinal (as in Kanamori?), Reinhardt cardinal (I know what Reinhardt’s axiom is, ), super Reinhardt cardinal Berkeley cardinal
Can someone step up here and explain these in generally understandable terms?
Peter’s message indicates how strong a grip Hugh has on , which if I recall, was offered as a “fix” for (inner model hypothesis) being incompatible with even an inaccessible cardinal.
Is merely a layering of the idea on top of large cardinal infrastructure?
I believe there has been repeated requests for a more substantial “fix” for (I think Hugh made such or similar requests). Which, if any, have been offered?
Are there any tangible prospects left for “fixing ” in order to shed light on CH? Are there any tangible prospects left for “fixing ” in order to shed light on any other mathematical open problem in set theory?
NOTE: I had concluded that, on the basis of this extensive traffic, this “HP” is not a legitimate foundational program, and should be renamed CTMP = countable transitive model program, a not uninteresting technical program that has been around for quite some time, with no artificial philosophical pretensions. Just the study of ctms. However, If Peter Koellner is going to put such an enormous amount of painstaking and skilled effort in continuing corresponding about it, I am more than happy to suspend disbelief and ask these questions.
NOTE: My own position is that “intrinsic maximality of the set theoretic universe” is prima facie a deeply flawed notion, fraught with nonrobustness (inconsistencies, particularly) that may or may not be coherently adjusted in order to lead to anything foundationally interesting. In my next message, I am hoping to play both sides of the fence. I will try my hand at coherently adjusting it. I will also try my hand at showing that the notion itself is inherently contradictory. I don’t know yet what I will find.