Tag Archives: Ideal V

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

Re: Paper and slides on indefiniteness of CH

Dear Pen,

On Sat, 18 Oct 2014, Penelope Maddy wrote:

Dear Sy,

With Claudio’s permission I will reply to your latest message (and also take the opportunity to make some further remarks).

I took Claudio to be proposing an interpretation of the HP program different from yours, one that’s an explicit multiverse program. No? If not, then I guess we return to the place where I still don’t quite grasp what the HPer is up to.  The universer is studying V. The HPer is studying …

Also V. But then in the HP a multiverse is used to analyse V. Recall my comment:

I don’t regard potentialism as multiversism.

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.

And indeed in the HP we have a “single-universe view” (the “ideal V”) which is analysed via a multiverse construct (the Hyperuniverse). It is a hybrid, and that may have caused confusion, for which I apologise. Another added ingredient is the consideration of “thickenings” in quotes, in addition to the lengthenings of the height potentialist. Both are reduced to the mathematical study of the Hyperuniverse (the reduction to the Hyperuniverse”, a bit more about that below).

… an ‘ideal V’?  What is that?

Just a colourful way to say “V, which we want to look as maximal as possible”. It is Max’s V.

The Hyperuniverse is also an ideal construct, it is defined within the ideal V. Conversely, truth in the ideal V is clarified through an analysis of its associated (ideal) Hyperuniverse. There is a dynamic interplay, a “dualism” as Claudio explained. There is no thick ontology for the ideal Hyperuniverse just as there is none for the ideal V.

The hyperuniverse is also ideal:  the collection of ctms inside ‘ideal V’, right?   What work is ‘ideal’ doing here?  Why not just V and the collection of ctms in V?

In my opinion, just to sound more colouful and to emphasize that we don’t fully grasp V, in its “ideal” fully maximal form (don’t theologians talk this way about not fully knowing God?). But Claudio may have somethong else in mind.

1. I can acknowledge that a single-universe view (with no multiverse consideratios) is the most obvious view for the foundations of Set Theory. But I don’t think it has better prospects than a multiverse view (or “hybrid” view as in the HP), since it has been quite obvious for a long time that independence is extensive in ST (Set Theory) and in my view this makes any single universe view that exceeds TR (Thin Realism) quite useless. What good is a single universe if we don’t know what form it takes and can only imagine a wide range of possibilities for that form?

Sure, the ordinary universer has a long way to go in figuring out the features of V.  But the aim remains a single theory of sets, which is the most natural way of satisfying the foundational goal.

Experience in ST shows that this is impossible with just Type 1 (set theory as a branch of math) evidence. Recall my assessment of CH: Type 1 undecidable, Type 2 (ST as a foundation) unknown ad Type 3 likely refutable.

My point is just that this goal generates a methodological maxim I once called ‘Unifiy':  go for one accepted theory of sets if at all possible.  (It might turn out to be impossible, given other goals, but it doesn’t seem to me that that has happened yet.)

I think “unify” is a great maxim, and it is also the basis for the HP theory of truth (unify maximality criteria). But in Type 1 it is not possible to achieve a complete unification, there will always be a collection of distinct theories. I think it likely that this will however happen in the HP.

What Claudio and I established was that, on his understanding of the HP, the HP has embraced the methodological maxim of Unify — not in the universer’s way, in a different way — so I was ready to move on to the next question: how to understand the ontology of the HP.

Claudio is right, the real HP work takes place in the Hyperuniverse (a multiverse) and “unify” is crucial for the analysis. The HP ontology is very analogous to that of the Thin Realist: There is a single universe V, but instead of it being determined by set-theoretic practice it is determined by the results of a Maximality Analysis which takes place in the Hyperuniverse. The other difference is that unlike set-theoretic practice, the Hyperuniverse is itself determined by V, so there is “feedback” between its own thin ontology and that of V.

As for the Thin Realist, she’s beholden to those extrinsic payoffs. Until they abound, she sticks to her simple universe understanding.

What “simple universe understanding”?

“HP identifies a core model-theoretic construct, that is, c.t.m., as the only constituent of multiverse [thin] ontology. Further, mathematical and logical, reasons for this choice have been explained at length by Sy, but I wish to recall that the main (and, to some extent, remarkable) fact is that we do not lose any information about set-theoretic truth by making this choice.”
I hope that this point has finally come across. Pen, Hugh and Harvey have each asked how I make the reduction to ctm’s and I can only ask them to please re-read what I have said about this at great length and through great effort in this exchange. The HP analyses Maximality through the study of certain very particular properties of ctm’s, but unlike what both Hugh and Harvey have tried to claim, it is much more than the study of ctm’s. The ctm’s are just the mathematical tool needed.

I realize this is frustrating for you, but what you’ve said so far about the ‘reduction to ctms’ hasn’t yet produced understanding (for me) or conviction (for Hugh or Harvey).  I’m still stuck, as above, on figuring out where the ctms live, what makes everything ‘ideal’, and so on.  They have other concerns.

I hope that this mail has helped clarify this.

What do you think of my “joining forces” idea? I do realise that TR and HP evidence are very different but one could embrace new axioms that are evidenced in both ways. I don’t know much about Ultimate L, for example, but if it proves to be good for ST-practice it would then be very interesting if it were a consequence of a compelling maximality criterion, something even Max could understand.

Best,
Sy

Re: Paper and slides on indefiniteness of CH

Dear Sy,

With Claudio’s permission I will reply to your latest message (and also take the opportunity to make some further remarks).

I took Claudio to be proposing an interpretation of the HP program different from yours, one that’s an explicit multiverse program. No?  If not, then I guess we return to the place where I still don’t quite grasp what the HPer is up to.  The universer is studying V. The HPer is studying …

And indeed in the HP we have a “single-universe view” (the “ideal V”) which is analysed via a multiverse construct (the Hyperuniverse). It is a hybrid, and that may have caused confusion, for which I apologise. Another added ingredient is the consideration of “thickenings” in quotes, in addition to the lengthenings of the height potentialist. Both are reduced to the mathematical study of the Hyperuniverse (the reduction to the Hyperuniverse”, a bit more about that below).

… an ‘ideal V’?  What is that?

The Hyperuniverse is also an ideal construct, it is defined within the ideal V. Conversely, truth in the ideal V is clarified through an analysis of its associated (ideal) Hyperuniverse. There is a dynamic interplay, a “dualism” as Claudio explained. There is no thick ontology for the ideal Hyperuniverse just as there is none for the ideal V.

The hyperuniverse is also ideal:  the collection of ctms inside ‘ideal V’, right?   What work is ‘ideal’ doing here?  Why not just V and the collection of ctms in V?

1. I can acknowledge that a single-universe view (with no multiverse considerations) is the most obvious view for the foundations of Set Theory. But I don’t think it has better prospects than a multiverse view (or “hybrid” view as in the HP), since it has been quite obvious for a long time that independence is extensive in ST (Set Theory) and in my view this makes any single universe view that exceeds TR (Thin Realism) quite useless. What good is a single universe if we don’t know what form it takes and can only imagine a wide range of possibilities for that form?

Sure, the ordinary universer has a long way to go in figuring out the features of V.  But the aim remains a single theory of sets, which is the most natural way of satisfying the foundational goal. My point is just that this goal generates a methodological maxim I once called ‘Unifiy':  go for one accepted theory of sets if at all possible.  (It might turn out to be impossible, given other goals, but it doesn’t seem to me that that has happened yet.)

What Claudio and I established was that, on his understanding of the HP, the HP has embraced the methodological maxim of Unify — not in the universer’s way, in a different way — so I was ready to move on to the next question:  how to understand the ontology of the HP.

As for the Thin Realist, she’s beholden to those extrinsic payoffs. Until they abound, she sticks to her simple universe understanding.

“HP identifies a core model-theoretic construct, that is, c.t.m., as the only constituent of multiverse [thin] ontology. Further, mathematical and logical, reasons for this choice have been explained at length by Sy, but I wish to recall that the main (and, to some extent, remarkable) fact is that we do not lose any information about set-theoretic truth by making this choice.”

I hope that this point has finally come across. Pen, Hugh and Harvey have each asked how I make the reduction to ctm’s and I can only ask them to please re-read what I have said about this at great length and through great effort in this exchange. The HP analyses Maximality through the study of certain very particular properties of ctm’s, but unlike what both Hugh and Harvey have tried to claim, it is much more than the study of ctm’s. The ctm’s are just the mathematical tool needed.

I realize this is frustrating for you, but what you’ve said so far about the ‘reduction to ctms’ hasn’t yet produced understanding (for me) or conviction (for Hugh or Harvey).  I’m still stuck, as above, on figuring out where the ctms live, what makes everything ‘ideal’, and so on. They have other concerns.

All best,
Pen