My participation in this interesting discussion is now at its end, as almost anything I say at this point would just be a repeat of what I’ve already said. I don’t regret having triggered this Great Debate on July 31, in response to your interesting paper, as I have learned enormously from it. Yet at the same time I wish to offer you an apology for the more than 500 subsequent e-mails, which surely far exceeds what you expected or wanted.
Before signing off I’d like to leave you with an abridged summary of my views and also give appropriate thanks to Pen, Geoffrey, Peter, Hugh, Neil and others for their insightful comments. Of course I am happy to discuss matters further with anyone, and indeed there is one question that I am keen to ask Geoffrey and Pen, but I’ll do that “privately” as I do think that this huge e-mail list is no longer the appropriate forum. My guess is that the vast majority of recipients of these messages are quite bored with the whole discussion but too polite to ask me to remove their name from the list.
All the best, and many thanks, Sy
1. Regarding Evidence for Set-Theoretic Truth and CH
I see three sources for such evidence: from set theory as a branch of mathematics (Type 1), from set theory as a foundation for mathematics (Type 2) and from set theory as a study of the set-concept (Type 3). More precisely, Type 1 evidence regards new axioms that best serve the interests of the development of “good set theory”, Type 2 evidence regards new axioms that best resolve independent questions in mathematics outside of set theory and Type 3 evidence regards axioms that are derivable from the maximality of V in height and width. In my view: There are and always will be competing Type 1 axioms, corresponding to the myriad forms of “good set theory”. Type 2 evidence has not yet been systematically explored, but at present the axioms that seem to do the best job are the Forcing Axioms, which have powerful combinatorial consequences of genuine interest to mathematicians outside of set theory. (Note: I am ignoring descriptive set theory, despite its deep and impressive connections with mathematics outside of set theory, because the part of descriptive set theory that is relevant to mathematics outside of set theory can be carried out in ZFC.) Type 3 evidence is only now being systematically explored via my Hyperuniverse Programme (HP).
Specifically regarding CH: I do not expect a resolution based solely on Type 1 evidence, as I expect that there will always be “good set theory” that implies it and “good set theory” that refutes it, due to the lack of consensus, even taking predications and verifications into account, about a single “good set theory”. However the very preliminary indications from Type 2 evidence are that the continuum has size and from Type 3 evidence that the continuum is very large, of size at least the first weakly inaccessible. This very preliminary evidence should however not be taken too seriously, as the studies of Type 2 and Type 3 evidence are in their infancy.
Finally, I would like to propose the view that the most desirable approach to truth in set theory is to combine all three of the above perspectives. If an axiom well serves the development of set theory as a branch of mathematics, is instrumental in resolving independent questions in mathematics outside of set theory and is compatible with (or better, derivable from) the maximality of V in height and width, then there is a strong case to be made for its truth. Of course meeting these three requirements (Types 1, 2 and 3 evidence all at the same time) is a very tall order and it is far too early to make any claims about what axioms may qualify for truth in this strong sense. But I do feel optimistic about our chances of success with this combined approach, perhaps not with CH but with other hypotheses, like PD or even large cardinals.
2. Regarding the HP (Hyperuniverse Programme)
Here I want to readily acknowledge the crucial contributions of participants in this discussion for forcing me to strengthen my own understanding of the HP. I began with a departure from the concept of set to the concept of set-theoretic universe, which after exchanges with Pen I chose to abandon. I also had “intrinsic features of V” in mind beyond just maximality, and thanks to Pen I simplified this to just “maximality of V in height and width”, agreeing to ground this form of maximality on intuitions shared by the set theory community. And in response to messages from Pen and Peter, I clarified that the HP aims to determine what is derivable from maximality in height and width and not what is “self-evident” or “unfoldable” from this feature; indeed this determination will be the result of a lengthy process which reaches a consensus, making heavy use of the mathematical techniques of set theory.
A point that caused confusion concerned the ontological framework. Motivated by a crucial message from Geoffrey, I chose to adopt a “single-multiverse” view enhanced by “height potentialism” (despite my own personal views which favour “radical potentialism”). In other words, there is a single V, but V can be lengthened and not thickened. I then explained how it is that one can nevertheless discuss properties of “thickenings” of V in terms of first-order properties of (mild) lengthenings of V and therefore implement forms of maximality which refer to “thickenings” (such as my original IMH criterion). A nice corollary of this is that one can apply Loewenheim-Skolem to argue that the analysis of the relevant maximality critieria can be undertaken entirely within the Hyperuniverse (the multiverse of countable transitive models of ZFC) without changing the results of that analysis. Although this reduction to the Hyperuniverse is optional, it offers a conceptually simpler way to express maximality criteria; thanks to Peter and Hugh I was forced to clarify the point that these Hyperuniverse criteria are expressed within a very restricted language and not all Hyperuniverse properties are expressible in this way.
That is where the HP stands at the moment. Thanks again mostly to Pen, but also to Geoffrey, Peter and Hugh for their helpful insights. There is a huge amount of work to be done concerning the formulation, analysis and unification of different maximality criteria. This is no easy task, as there are many plausible candidates for criteria which reflect the maximality of V in height and width. Moreover some maximality criteria are seen to be inconsistent only after hard mathematical work. I ask my colleagues not to judge the programme before seeing the maximality criteria that it develops as the result of a careful analysis. You will see that valuable evidence for set-theoretic truth will result.