Re: Paper and slides on indefiniteness of CH

Dear Pen and Harvey,

Sorry for not replying sooner to your questions.

I believe I should address some of the issues that I had left unanswered and maybe provide some further responses to the questions you’d asked. Wrt to Harvey’s, I think that Radek has already given some persuasive answers, so I will for the time being concentrate on Pen’s questions.

Sy, please feel free at any point to add any remarks you may find useful to contribute to the discussion.

Pen, in a sense you’re right, the hyperuniverse “lives” within V (I’d rather say that it “originates from” V) and my multiverser surely has a notion of V as anybody else working with ZFC. Moreover, members of H are thought to be strongly related to V also in another way, through the satisfaction of principles which are, originally, assumed to be referring to the universe. By setting up the hyperuniverse concept and framework, however, she stipulates that questions of truth about V be dealt within the hyperuniverse itself. Again, this doesn’t imply that whatever is taken to hold across portions of the hyperuniverse is then referred back to the universe.

I’m not sure that Sy entirely agrees with me on this point, but to me HP implies an irreversible departing from the idea of finding a single, unified body of set-theoretic truths. Even if a convergence of consequences of H-axioms were to manifest itself in a stronger and more tangible way, via, e.g., results of the calibre of those already found by Sy and Radek, I’d be reluctant to accept the idea that this would automatically reinstate our confidence in a universe-view through simply referring back such a convergence to a pristine V.

Moreover, HP, in my view, constitutes the reversal of the foundational perspective I described above (that is, to find an ultimate universe), by deliberately using V as a mere inspirational concept for formulating new set-theoretic hypotheses rather than as a fixed entity whose properties will come to be known gradually. As has been indicated by someone in this thread, HP fosters an essentially top-down approach to set-theoretic truth, whose goal is that of investigating what truth about sets may be generated beyond that incapsulated by ZFC using new information about V. I used the term ideal in my previous email just to convey the contrast between a V investigated through attributing to it certain features and a real V as a fixed entity progressively determined through subsequent refinements (I guess that this ideal status of V might be construed in the light of Kant’s Grenzbegriffe working as regulatory ideals [I owe this interpretation of V to Tatiana Arrigoni]).

Coming to the other issues raised, c.t.m. have proved to be technically very expedient and fertile in terms of consequences for the purposes of HP and, moreover, they seem to capture adequately the basic intuitions at work in such techniques as forcing. And as you pointed out, the hyperuniverse might be seen as something allowing us to generate a unified conceptual arena to study a multiverse framework, so why wouldn’t the use of c.t.m. be justified precisely on these specific grounds?

To go back to a more general point, I believe that HP should be judged essentially for its merits as a dynamic interpretation of truth within a multiverse framework. In my view, its construction, thus, responds, to a legitimate foundational goal, provided one construes foundations not in the sense of selecting uniquely and determinately the best possible general axioms for the mathematics we know (including set-theoretic mathematics), but rather in that of exhibiting (and studying) evidential processes for their selection: the study of what I defined properties of an ideal V within the hyperuniverse is one such evidential process.

Now, as I said in my previous email, the programme, at least as far as its epistemological goals are concerned, is far from being perfected in all its parts, of course. As said (and requested by many people also here), it has to clarify, for instance, what further legitimate ideal properties of V there are, and what justifies their probable intrinsicness (relationship to the set concept).

Has this brief summary answered (at least some of) your legitimate concerns?

Best wishes,
Claudio

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