Re: Paper and slides on indefiniteness of CH

Dear Sy,

 Type 2 comes down HARD for Forcing Axioms and V = L, as so far none of the others has done anything important for mathematics outside of set theory.

I was assuming that any theory capable of ‘swamping’ all others would ‘subsume’ the (Type 1 and Type 2) virtues of the others.  It has been argued that a theory with large cardinals can subsume the virtues of V=L by regarding them as virtues of working within L.  I can’t speak to forcing axioms, but I think Hugh said something about this at some point in this long discussion.

All best,

Pen

Re: Paper and slides on indefiniteness of CH

Dear Sy,

Peter is right, Sy. There’s no difference of opinion here between Peter and me about what counts as evidence, whether we call it ‘good set theory’ or ‘P and Vs’.

There is another point. Wouldn’t you want a discussion of truth in set theory to be receptive to what is going on in the rest of mathematics?

I don’t mean to be cranky about this, Sy, but I’ve lost track of how many times I’ve repeated that my Thin Realist recognizes evidence of both your Type 1 (from set theory) and Type 2 (from mathematics). I think I’ve mentioned that the foundational goal of set theory in particular plays a central role (especially in Naturalism in Mathematics).

All best,

Pen

Re: Paper and slides on indefiniteness of CH

Dear Sy,

You wrote:

When it comes to Type 1 evidence (from the practice of set theory as mathematics) we don’t require that opinions about what is “good set theory” be shared (and “the picture” is indeed determined by “good set theory”). As Peter put it:

” Different people have different views of what “good set theory” amounts to. There’s little intersubjective agreement. In my view, such a vague notion has no place in a foundational enterprise.”

What Peter wrote is this:

The notion of “good set theory” is too vague to do much work here. Different people have different views of what “good set theory” amounts to. There’s little intersubjective agreement. In my view, such a vague notion has no place in a foundational enterprise. The key notion is evidence, evidence of a form that people can agree on.

I probably should have stepped in at the time to remark that I’ve been using the term ‘good set theory’ for the set theory that enjoys the sort of evidence Peter is referring to here — for example, there was evidence for the existence of sets in the successes of Cantor and Dedekind, and more recently for PD, in the results cited by Peter, Hugh, John Steel, and others.  (Using the term ‘good set theory’ allows me to leave open the question of Thin Realism vs. Arealism.  For the Arealist, these same considerations are just reasons to add sets or PD to our mathematics/set theory, but the Thin Realist sees them as evidence for existence and truth.)

This doesn’t preclude people disagreeing about what parts of set theory they believe to be more interesting, important, promising, etc.  (Scientists also disagree on such matters.)

At the present juncture, it’s more difficult to find and assess new evidence, but that’s to be expected.  Peter and Hugh have made it clear, I think, that they regard many of the current options as open (including HP, when it begins to generate definite claims), that more information is needed. If one theory eventually ‘swamps’ the rest (I should have noted that ‘swamping’ often involves ‘subsuming’), then the apparently contrary evidence will have to be faced and explained.  (Einstein had to explain why there was so much evidence in support of Newton.)

All best,

Pen

Re: Paper and slides on indefiniteness of CH

Dear Sy,

Pen and Peter, can you please help here? Pen hit me very hard for developing what could be regarded as “Sy’s personal theory of truth” and it seems to me that we now have “Hugh’s personal theory of truth”, i.e., when Hugh develops a powerful piece of set theory he wants to declare it as “true” and wants us all to believe that. This goes far beyond Thin Realism, it goes to what Hugh calls a “conception of V” which far exceeds what you can read off from set-theoretic practice in its many different forms. Another example of this is Hugh’s claim that large cardinal existence is justified by large cardinal consistency; what notion of “truth” is this, if not “Hugh’s personal theory of truth”?

Pen’s Thin Realism provides a grounding for Type 1 truth. Mathematical practice outside of set theory provides a grounding for Type 2 truth. Out intuitions about the maximality of V in height and width provide a grounding for Type 3 truth. How is Hugh’s personal theory of truth grounded?

I’m pretty sure Hugh would disagree with what I’m about to say, which naturally gives me pause. With that understood, I confess that from where I sit as a relatively untutored observer, it looks as if the evidence Hugh is offering is overwhelming of your Type 1 (involving the mathematical virtues of the attendant set theory). My guess is he’d also consider type 2 evidence (involving the relations of set theory to the rest of mathematics) if there were some ready to hand. He has a ‘picture’ of what the set theoretic universe is like, a picture that guides his thinking, but he doesn’t expect the rest of us to share that picture and doesn’t appeal to it as a way of supporting his claims. If the mathematics goes this way rather than that, he’s quite ready to jettison a given picture and look for another. In fact, at times it seems he has several such pictures in play, interrelated by a complex system of implications (if this conjecture goes this way, the universe like this; if it goes that way, it looks like that…) But all this picturing is only heuristic, only an aide to thought — the evidence he cites is mathematical. And, yes, this is more or less how one would expect a good Thin Realist to behave (one more time: the Thin Realist also recognizes Type 2 evidence). (My apologies, Hugh. You must be thinking, with friends like these…)

The HP works quite differently. There the picture leads the way — the only legitimate evidence is Type 3. As we’ve determined over the months, in this case the picture involved has to be shared, so that it won’t degenerate into ‘Sy’s truth’. So far, to be honest, I’m still not clear on the HP picture, either in its height potentialist/width actualist form or its full multiverse form. Maybe Peter is doing better than I am on that.

All best,

Pen

Re: Paper and slides on indefiniteness of CH

Dear Harvey,

I’d like to add one more general observation to these [Koellner's] compelling remarks. In cases like this one, where the evidence is overwhelmingly extrinsic, there’s no reason to expect the relevant discussions to be ‘generally understandable’. It’s only in cases like the HP, where the evidence is intended to be primarily or even exclusively intrinsic, that ‘general understandability’ becomes crucial. (If the principles are supposed to follow from the concept of set, then we need to be able to see how that works.)

All best,

Pen

PS to Peter: Thank you very much for your message and slides about the current state of understanding on choiceless cardinals! I look forward to spending a bit more time with them both.

Re: Paper and slides on indefiniteness of CH

Dear Sy,

Let’s first note that in the wake of independence, it’s going to be a pretty hard-line Universist (read “nutty Universist”) who asserts that we shouldn’t be studying truth across models in order to understand V better.

Pen Maddy is not a nut! You can simply ground truth in what is good set theory and mathematics, as would a Thin Realist (right, Pen?) and not bother with all of this talk about models.

This doesn’t really bear on what you and Neil are discussing, but for the record, the Thin Realist will study models of set theory as readily as the next set theorist.  Such a study might well turn out some good set theory and/or good mathematics more generally (which is how she comes ‘to understand V better’).  And, as remarked earlier, she doesn’t regard herself as infallible:  her commitment to a single universe could change in light of further developments.

All best,

Pen

Re: Paper and slides on indefiniteness of CH

Dear Sy,

We already know why CH doesn’t have a determinate truth value, it is because there are and always will be axioms which generate good set theory which imply CH and others which imply not-CH. Isn’t this clear when one looks at what’s been going on in set theory?

Well, I’m not sure it is clear that there will never be a theory whose virtues swamp the rest.

 is CH one of the leading open questions of set theory?

No! The main reason is that, as Sol has pointed out, it is not a mathematical problem but a logical one. The leading open questions of set theory are mathematical.

I didn’t realize that you’d been convinced by Sol’s arguments here.  My impression was that you thought it  might be possible to resolve CH mathematically:

 I started by telling Sol that the HP might give a definitive refutation of CH! You told me that it’s OK to change my mind as long as I admit it, and I admit it now!

That’s why I posed the question to you as I did.

All best,
Pen

Re: Paper and slides on indefiniteness of CH

Dear Sy,

If we are talking about ST in terms of its role as a foundation for or subfield of mathematics (Types 1 and 2) then we needn’t trouble ourvselves with this discussion of universes for ST and can hang our hats on what axioms of set theory are advantageous for the development of set theory and mathematics, as was done with AC and the Axiom of Infinity, for example.

Thanks for this clarification.  If all we care about is set theory as a branch of mathematics and set theory as it relates to the rest of mathematics, then we can stick with our familiar iterative picture of V and rely on extrinsic justifications of the familiar sort (unless the extrinsic evidence eventually leads us to prefer some sort of multiverse, in which case we’d shift to a new picture).  It’s only when we’re interested in further exploration of ‘the maximality of the set-concept’ that we need to engage in the HP (or the MP).

All best,
Pen

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 Sy,

This doesn’t really bear on any of the debates we’ve been having, but …

It seems to me disingenuous to suggest that resolving CH, and devising a full account of sets of reals more generally, is not one of the goals of set theory — indeed a contemporary goal with strong roots in the history of the subject.

Good luck selling that to the ST public. This is interesting to you, me and many others in this thread, but very few set-theorists think it’s worth spending much time on it, let’s not deceive ourselves. They are focused on “real set theory”, mathematical developments, and don’t take these philosophical discussions very seriously. … Resolving CH was certainly never my goal; I got into the HP to better understand large cardinals and internal consistency, with no particular focus on CH. … It would be interesting to ask other set-theorists (not Hugh or I) what the goals of set theory are; I think you might be very surprised by what you hear, and also surprised by your failure to hear “solve CH”.

The goal I mentioned was resolving CH as part of a full theory of sets of reals more generally. I said ‘resolving’ to leave open the possibility that the ‘resolution’ will be a understanding of why CH doesn’t have a determinate truth value, after all (e.g., a multiverse resolution).

It’s not a matter of how many people are actively engaged in the project: there might be lots of perfectly good reasons why most set theorists aren’t (because there are other exciting new projects and goals, because CH has been around for a long time and looks extremely hard to crack, etc.). I would ask you this: is CH one of the leading open questions of set theory? Is it the sort of thing that would draw great acclaim if someone were to come up with a widely persuasive ‘resolution’?

All best,
Pen