Tag Archives: Multiverse conception

Re: Paper and slides on indefiniteness of CH

Dear Geoffrey,

I’d have thought that a true “multiverser” would want to replace all talk of “V” –understood as the universe of [absolutely] all ordinals(and sets, etc.)–with some more  benign term, such as some very large, (perhaps maximally) fat, transitive model of (here a ref to ZFC + some very large card axioms).

I was imagining the multiverser saying that there isn’t just one universe, there are a bunch — with some account to tell us what universes there are and what they’re like.  None of them is V.

But, for most mathematical purposes outside the higher reaches of set theory itself, I thought that it wouldn’t matter. Several messages back, ref was made to how a group theorist, for instance, might choose. But couldn’t either view accommodate any new axiom that might possibly matter to a group theorist? That seems to be the case with my modal version of multiverse, in which the possible structures are, up to isomorphism, linearly ordered by “end-extension”.

This is why I asked Claudio if a potentialist (like you) counts as a multiverser.  In practice, it doesn’t seem there’s a lot of difference between your potentialist multiverser and a universer who says:  there’s a single fixed universe, but we can’t describe it completely; we have to keep adding more large cardinal axioms.  If the algebraist comes to the set theorist in his foundational role and asks a question turns out to hinge on, say, inaccessibles (as apparently in Wiles’ original proof), you’d say, ‘no problem, what you want lives in this end-extention’, and my universer would say, ‘no problem, there are inaccessibles’.

But I was imagining that Claudio’s multiverse would be more varied that that.  So I floated a couple of possibilities:

You might say to the algebraist:  there’s a so-and-so if there’s one in one of the universes of the multiverse.  Or you might say to the universer that her worries are misplaced, that your multiverse view is out to settle on a single preferred theory of sets, it’s just that you don’t think of it as the theory of a single universe; rather, it’s somehow suggested by or extracted from the multiverse.

(Claudio seemed to opt for the second, but ultimately rejected it; I’m not sure what he thinks about the first.)

On the first, for our simple example, the multiverser would presumably say pretty much what your potentialist says:  here, work in this universe with inaccessibles.  For this to work, the multiverser would need to give us a theory of what universes there are.  For your simple height potentialist, perhaps we have this, but the more varied multiverser would owe us such an account.  (I would have asked about that if Claudio had gone for this option.)

Matters get a little harder when the algebraist is after something dicier.  Suppose he wants a definable (projective) well-ordering of the reals.  My universer might say:  well, there’s isn’t such a thing, but if you restrict yourself to thinking inside L, you can have one there; just be sure that all the other apparatus you need is available there, too.  Would your potentialist want to say something like that?

All best,

Pen

PS:  To be honest, I have the uneasy feeling that there’s something off in this way of thinking about the foundational goal, but I don’t know what it is.

Re: Paper and slides on indefiniteness of CH

Dear Pen,

On Oct 19, 2014, at 6:36 PM, Penelope Maddy wrote:

Dear Claudio,

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.

OK, but now I lose track of the sense in which yours is a multiverse view: there’s V and within V there’s the hyperuniverse (the collection of ctms). Any universer can say as much.

I understand my language was ambiguous. I wasn’t claiming that the whole hyperuniverse is within V. That is simply impossible, insofar as there are members of H which satisfy CH and others which don’t, some which satisfy IMH and some which don’t and so on. However, it is always possible (and logically necessary) to see any member of H as living in V. Any multiverser may concede that universes, say, mutually differring set-generic models, are in V, but this doesn’t commit her to be a universer.

I take the universer to believe that there’s one single (or definitive) V we perceive approximately and incompletely and/or we may gradually determine uniquely, whereas I take the multiverser to simply affirm that V is a collection of different set-theoretic universes each of which is endowed with differring properties. That is why a multiverser cannot pursue unification: universes contain information which cannot be amalgamated into one single universe (an ultimate V). We may say that V is, for the multiverser, an archetypal universe, not a unique or unifying reality.

Now, HP thinks that an amalgamation (via the convergence phenomenon) might be, in principle, possible, but my construal of it is that, even at that point, there’d be no need to go back to a universe-view.

(I confess this is a disappointing. I was hoping that a true multiverser would be joining this discussion.)

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.

Now I’m confused again. Here’s the formulation you agreed to:

Or you might say to the universer that her worries are misplaced, that your multiverse view is out to settle on a single preferred theory of sets, it’s just that you don’t think of it as the theory of a single universe; rather, it’s somehow suggested by or extracted from the multiverse.

Though embracing a single universe is the most straightforward way of pursing unify, I was taking you to be pursuing it in a multiverse context (not to be embracing ‘a pristine V’). Fine with me.

But now that you’ve clarified that you aren’t really a multiverser, that you see all this as taking place within V, why reject unify now? And if you do, what will you say to our algebraist?

See above. I believe HP is a multiverse theory, therefore an HPer ought not to foster unification (but rather justified selection of universes). I’m not sure that all HP people would agree on this, though.

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.

So there’s a sense in which you have V and a sense in which you don’t. If V is so indeterminate, how can the collection of ctms within it be a well-defined object open to precise mathematical investigation?

In fact, it seems to me that practice shows us that a c.t.m. is easier to deal with and, in the end, more informative than real V (incidentally, as also pointed out by Radek, that is the key to understanding forcing).

Best wishes,
Claudio

Re: Paper and slides on indefiniteness of CH

Dear Claudio,

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.

OK, but now I lose track of the sense in which yours is a multiverse view: there’s V and within V there’s the hyperuniverse (the collection of ctms). Any universer can say as much.

(I confess this is a disappointing. I was hoping that a true multiverser would be joining this discussion.)

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.

Now I’m confused again. Here’s the formulation you agreed to:

Or you might say to the universer that her worries are misplaced, that your multiverse view is out to settle on a single preferred theory of sets, it’s just that you don’t think of it as the theory of a single universe; rather, it’s somehow suggested by or extracted from the multiverse.

Though embracing a single universe is the most straightforward way of pursing unify, I was taking you to be pursuing it in a multiverse context (not to be embracing ‘a pristine V’). Fine with me.

But now that you’ve clarified that you aren’t really a multiverser, that you see all this as taking place within V, why reject unify now? And if you do, what will you say to our algebraist?

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.

So there’s a sense in which you have V and a sense in which you don’t. If V is so indeterminate, how can the collection of ctms within it be a well-defined object open to precise mathematical investigation?

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

I very much appreciate your efforts, Claudio but the picture still isn’t clear to me. A simple, readily understandable intuitive picture can be an immensely fruitful tool, as the iterative conception has amply demonstrated, but this one, the intuitive picture behind the HP, continues to elude me.

All best,

Pen

Re: Paper and slides on indefiniteness of CH

Dear Pen,

If I may make a couple of remarks (with apologies for intervening):

On Sun, 19 Oct 2014, Penelope Maddy wrote:

Dear Claudio,

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 workin g with ZFC.

OK, but now I lose track of the sense in which yours is a multiverse view: there’s V and within V there’s the hyperuniverse (the collection of ctms). Any universer can say as much.

Yes, but what is new is to use a multiverse as a tool to gain knowledge about V. Of course the hyperuniverse is very “multi” and not “single” in its form.

(I confess this is a disappointing. I was hoping that a true multiverser would be joining this discussion.)

I am a radical multiverser but have agreed to suppress that because you and Geoffrey didn’t like width potentialism and switching to your view (multiverser “light”: height potentialism only) doesn’t cost the programme much.

I’m not sure that Sy entirely agrees with me on this point, but to me HP implies an irre versible departing from the idea of finding a single, unified body of set-theoretic trut hs. Even if a convergence of consequences of H-axioms were to manifest itself in a str onger 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 reinsta te our confidence in a universe-view through simply referring back such a convergence to a pristine V.

Now I’m confused again. Here’s the formulation you agreed to:

Or you might say to the universer that her worries are misplaced, that your multiverse view is out to settle on a single preferred theory of sets, it’s just that you don’t think of it as the theory of a single universe; rather, it’s somehow suggested by or extracted from the multiverse.

Though embracing a single universe is the most straightforward way of pursing unify, I was taking you to be pursuing it in a multiverse context (not to be embracing ‘a pristine V’). Fine with me.

“Unify” plays a huge role in the Hyperuniverse analysis. I called it “synthesis” before. It is only with “Unification” that one gets convergence towards a single theory of truth in the HP.

But now that you’ve clarified that you aren’t really a multiverser, that you see all this as taking place within V, why reject unify now? And if you do, what will you say to our algebraist?

Moreover, HP, in my view, constitutes the reversal of the foundational perspective I des cribed 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.

So there’s a sense in which you have V and a sense in which you don’t. If V is so indeterminate, how can the collection of ctms within it be a well-defined object open to precise mathematical investigation?

It’s remarkable what one can learn working with the Hyperuniverse construct, even though it is as “undetermined” as V itself. Don’t forget the “feedback” feature: Working with the Hyperuniverse clarifies what V is, and that in turn has some effect (admittedly less dramatic) on what the Hyperuniverse is. Another key point is that the Hyperuniverse, due to the countability of its members, facilitates an enormous wealth of techniques not available for uncountable universes.

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

I very much appreciate your efforts, Claudio but the picture still isn’t clear to me. A simple, readily understandable intuitive picture can be an immensely fruitful tool, as the iterative conception has amply demonstrated, but this one, the intuitive picture behind the HP, continues to elude me.

I’ll not interfere with Claudio’s response to this. Indeed, his views and mine are not exactly the same. Radek and Tatiana also have their own distinctive points of view, even though they also support the HP in some way. I am grateful to both Claudio and Radek for joining the discussion (even though I didn’t ask them to!).

Best,
Sy