Sy wrote:

In other words, we can discuss lengthenings and shortenings of V without declaring ourselves to be multiversers. Similarly we can discuss “thickenings” in quotes. No multiverse yet. But then via a Loewenheim-Skolem argument we realise that it suffices to work with a countable little-V, where it is natural and mathematically extremely useful to regard lengthenings and “thickenings” as additional universes. Thus the reduction of the study of Maximality of V to the study of mathematical criteria for the selection of preferred “pictures of V” inside the Hyperuniverse, The Hyperuniverse is of course entirely dependent on V; if we accept a new axiom about V then this will affect the Hyperuniverse. For example if we accept a little more than first-order reflection then a consequence is that the Hyperuniverse is nonempty.

If Sy would slow down and carefully explain in universally understandable terms just what he is talking about, we would all probably recognize that the use of the “Löwenheim-Skolem argument” is bogus.

I’m not sure that Sy is aware that there are some standards for doing philosophy and foundations of set theory (or anything else). Perhaps Sy believes that with enough energetic offerings of slogans, and enough seeking of soundbites from philosophers (which he has found are not all that easy to get), you can avoid having to come up with real foundational/philosophical ideas that work.

I am not aware of a single person on this email list who is inclined to believe that CTMP (aka HP) constitutes any kind of legitimate foundational program for set theory – at least on the basis of anything offered up here. (CTMP appears to be a not uninteresting technical study, but even as a technical study, it currently suffers from a lack of systemization – at least judging by what is being offered up here).

If there is a single person on this email list who thinks that CTMP (aka HP) constitutes any kind of legitimate foundational program for set theory, I think that we would all very much appreciate that they come forward and say why they think so, and start offering up some clear, deliberate, and generally understandable answers to the questions I raised a short time ago. I copy them below.

Now I am not primarily here to tear down silly propoganda. Enough of this has already been done by me and others. I am making efforts to steer this discussion into productive channels that meet that great standard: being generally understandable to everybody, with no attempt to mask flawed ideas — or seemingly unsound ideas — in a mixture of technicalities, slogans, and propoganda I invited Sy to engage in a productive discussion that would meet at least minimal standards for how foundations and philosophy can be discussed, and he has refused to engage dozens of times.

So again, if there is anybody here who thinks that CTMP (aka HP) is a legitimate foundational program for set theory, please say so, and engage in the following questions I posted recently: In the meantime, I am finishing up a wholly positive message that I hope you are interested in.

**QUESTIONS** – lightly edited from the original list

Why doesn’t HP carry the obvious name CTMP = countable transitive model program. That is my suggestion and has been supported by Hugh.

What does the choice of a countable transitive model have to do with “(intrinsic) maximality in set theory”? Avoid quoting complicated technicalities, meaningless slogans, or idiosyncratic jargon and adhere to generally understandable considerations.

At a fundamental level, what does “(intrinsic) maximality in set theory” mean in the first place?

Which axioms of ZFC are motivated or associated with “(intrinsic) maximality in set theory”? And why? Which ones aren’t and why?

What is your preferred precise formulation of IMH? E.g., is it in terms of countable models?

What do you make of the fact that the IMH is inconsistent with even an inaccessible (if I remember correctly)? If this means that IMH needs to be redesigned, how does this reflect on whether CTMP = HP is really being properly motivated by “(intrinsic) maximality in set theory”?

What is the simplest essence of the ideas surrounding “fixing or redesigning IMH”? Please, in generally understandable terms here, so that people can get to the essence of the matter, and not have it clouded by technicalities.

Overall, it would be particularly useful to avoid quoting complicated technicalities or idiosyncratic jargon and adhere to generally understandable considerations. After all, CTMP = HP is being offered as some sort of truly foundational program. Legitimate foundational programs lend themselves to generally understandable explanations with overwhelmingly attractive features.

Harvey