# Re: Paper and slides on indefiniteness of CH

Dear Peter,

Before I forget I should mention now that tomorrow I will be off to a conference, so may not be responding to e-mail promptly for the rest of this week.

Thanks a lot for your messages; they are very helpful for sharpening the arguments that I am making. And I apologise if my description of radical potentialism caused so much confusion! Let me try to clarify it better in this mail.

Your second point is about “desirable properties”; let me address that first. The HP is aimed primarily at what is derivable from the intrinsic feature of maximality, i.e. it is concerned with the maximal iterative conception. But I also mentioned “omniscience”, which I do not see as derivable from maximality, at least no one has presented such an argument and I don’t know of one. If omniscience were to be included as intrinsic to the “concept of set” then Pen would have been right to say that we have changed to a different concept! I only used the word “desirable feature” informally to suggest that I find omniscience desirable, nothing more. I would very much like to hear suggestions about what practice-independent notions like omniscience should be regarded as “desriable”; I have no idea how to formulate that. Actually I am curious to know: Do you see it as a “desirable” feature of the universe of sets? Maybe you don’t want to talk about “desirable features” at all, and I can understand that.

OK, now to radical potentialism: Maybe it would help to talk first about something less radical: Width potentialism. In this any picture of the universe can be thickened, keeping the same ordinals, even to the extent of making ordinals countable. So for any ordinal alpha of V we can imagine how to thicken V to a universe where alpha is countable. So any ordinal is “potentially countable”. But that does not mean that every ordinal *is* countable! There is a big difference between universes that we can imagine (where our aleph_1 becomes countable) and universes we can “produce”. So this “potential countability” does not threaten the truth of the powerset axiom in V!

The standard form of potentialism can be viewed as a process of lengthening as opposed to thickening. Once again, there is no model of ZFC “at the end” because there is no “end”.

Now radical potentialism is in effect a unification of these two forms of potentialism. We allow V to be lengthened and thickened simultaneously. If we were to keep thickening to make every ordinal of V countable then after $\text{Ord}(V)$ steps we are forced to also lengthen to reach a (picture of a) universe that satisfies ZFC. In that universe, the original V looks countable. But then we could repeat the process with this new universe until it is seen to be countable. The potentialist aspect is that we cannot end this process by taking the union of all of our pictures. In fact, whereas in the standard discussion of lengthenings there could be a debate about whether we can arrive at “the end”, if we allow both lengthenings and thickenings, potentialism is the only possibility; actualism is ruled out because the union of our “universes” would not be a model of ZFC and would therefore have to be lengthened further! And again, the “potential countability of V” does not threaten the truth of the axioms of ZFC in V!

Now in powerset and ordinal maximality we are not comparing V to pictures of other universes which see V as countable, even though there are such pictures. We are only looking at lengthenings that have V as a rank-initial segment and thickenings that have the same ordinals as V. From the perspective of a given V, these lengthenings and thickenings are only pictures of course, we are not talking about actual universes of sets, as those would be contained in V. But as I said in my last mail, even a platonist, with his own special V can imagine lengthenings and thickenings. It seems that I have some platonistically-leaning colleagues who discuss the set-generic multiverse surrounding V, which makes no sense if all universes are contained in V. The relevant set-generic extensions can be “pictured” but not “produced”. There are other constructions which take a countable universe and lengthen it, and doing this to V can also be “pictured” by a Platonist.

So set theory has not evaporated, CH is still a good problem. There is a huge wealth of pictures of V and some are “better” than others in the sense that some are better witnesses to maximality than others. The minimal model of ZFC is a terrible witness to maximality. A witness to the IMH does a much better job. In the HP we want to figure out which are the “best” witnesses to maximality. We may conclude that these “best” witnesses to maximality satisfy not CH, or we may conclude otherwise. It is too early to make such a judgment.

Now the next move (please see the outline) is to realise that if the spectrum of pictures is as rich as I describe, allowing V to “look countable” in pictures of larger universes, then in the “maximality test” where V is compared to pictures obtained through thickening and lengthening we might as well carry out this test inside a large picture of V where the orignal universe looks countable and where the lengthenings and thickenings you need actually exist as transitive models of ZFC. The result is that if you want to know if something first order holds in all universes that pass the “maximality test” you can simply assume that the test is taking place in the Hyperuniverse of some background V. (Of course depending on the choice of that backgound V, there may or may not exist universes that pass the maximality test.) This is the reduction of the problem to the Hyperuniverse, where these pictures can actually be realised as transitive models of ZFC.

Best,
Sy

# Re: Paper and slides on indefiniteness of CH

Dear Sol,

This message is not specifically about your rebuttal of Hilary’s claim, but about a more general issue which I hope that you can shed light on.

You write:

Q1. Just which mathematical entities are indispensable to current scientific theories?, and
Q2. Just what principles concerning those entities are need for the required mathematics?

My very general question is: What do we hope to gain by showing that something can be “captured by limited means” (in this case regarding what mathematics is needed for physical theory)? Does this tell us something new about what we have “captured”?

I am of course familiar with advantages of, for example, establishing that some computable function is in fact provably total in PA, as then one might extract useful and new information about the growth rate of such a function. In set theory is something analagous, which is if you can bring down the large cardinal strength enough, core model theory kicks in and you have a good chance of achieving a much better understanding. Or if one starts with a philosophical position, like predicativity, it is somehow gratifying to know that one can capture it precisely with formal means.

But frankly speaking, too often there is a connotation of “of, we don’t really need all of that bad set theory to do this”, or even more outdated: “what a relief, now we know that this is consistent because we captured it in a system conservative over PA!”. Surely in the 21st century we are not going to worry anymore about the consistency of ZFC.

Is the point that (as you say at the end of your message) that you think you have to invoke some kind of platonistic ontology if you are not using limited means, and for some reason this makes you feel uncomfortable (even though I presume you don’t have inconsistency worries)?

It is tempting to think that your result using your system W might tell us something new about physics. Does it? On the other hand you have not claimed that “physics is conservative over PA” exactly, but only that the math needed to do a certain amount of physics is conservative over PA.

Finally, how is it that you claim that “only a platonistic philosophy of mathematics provides justification” for impredicative 2nd order arithmetic? That just seems wrong, as there are plenty of non-platonists out there (I am one) who are quite happy with ZFC. But maybe I don’t understand how you are using the word “justification”.

Thanks in advance for your clarifications. And please understand, I am not suggesting that it is not valuable to “capture things by limited means”, I just want to have a better understanding of what you feel is gained by doing that.

All the best,
Sy

# Re: Paper and slides on indefiniteness of CH

Dear Geoffrey and Sol,

The issue is whether “the totality of all full models of third-order number theory” is a legitimate vehicle for establishing that statements like CH have a definite truth value.

I was trying to shake Geoffrey’s supreme confidence that CH has a definite truth value by appealing to statements outside set theory that clearly do not have a definite truth value.
So that’s why we went outside math into science, and also to

1. Mozart is a better musician than Beethoven.
2. Geoffrey is bald.

All that we have about sets of integers is that they are extensions of ARBITRARY predicates on the integers. That’s not much of an explanation. It does tell us that if we come across any way of giving a predicate then we can form the set. The full impredicative comprehension axiom scheme looks very compelling – although it can be criticized in. Nothing like full comprehension, even on the reals, has the power to decide CH. Any concept of “set theoretic universe” makes for a far far more complex story. The notion of arbitrary ordinal looks much harder to argue is definite in the appropriate sense that even “set of reals”.

Experience shows that almost no one who has thought about these matters has ever changed their mind based on an argument even as well thought out as Sol’s. So the kind of interchange Geoffrey and I have been having does seem to be relevant.

I tend to think that we are all spoiled by just how amazingly far we can go in the finite with very minimal principles, and also even just how far ZFC goes. We think that if we can go so amazingly far, then it is because of some underlying Platonic reality, or at least a way of doing set theory that is unequivocally forced on us. It seems likely that we are mistaken.

But you still have to tell a convincing story. For skeptics, the story is only interesting to the extent that is simple and compelling.

Again, I never see anybody change their minds on the “definite truth value” issue. But there is a place where maybe we can agree. That is on the issue of whether CH research is a relatively promising area of research in the foundations of mathematics. I read Solovay as saying “no”, even though he a “committed Platonist”. What does Geoffrey think?

Harvey

# Re: Paper and slides on indefiniteness of CH

This is in response to the following quote of Harvey:

Incidentally, do you agree with me that CH research is not a relatively promising area of f.o.m. research? I tend to believe that people think CH research is a promising area of f.o.m. research if and only if they subscribe to “CH has a determinate truth value”.

I am a convinced platonist and fully subscribe to the proposition that CH has a definite truth value.

I am quite sceptical about the prospects of determining CH by any approach currently on the horizon. In particular, I doubt that anything growing out of the work of Sy Friedman or any of the work of Woodin, past or present, that I know about will lead to any determination that I find in the least convincing.

I have enormous respect for Woodin’s mathematical achievements, even though I do not think they have any prospect of leading to a solution of the continuum problem.

Bob Solovay

# Re: Paper and slides on indefiniteness of CH

Dear Sy,

There is no retreat from my view that the concept of the continuum (qua the set of arbitrary subsets of the natural numbers) is an inherently vague or indefinite one, since any attempt to make it definite (e.g. via L or an L-like inner model) runs counter to what it is supposed to be about. I talk here about the concept of the continuum, not the supposed continuum itself, as a confirmed anti-platonist.  Mathematics in my view is about intersubjectively shared (human) conceptions of idealized structures, not any supposed such structures in and of themselves.  See my article “Conceptions of the continuum” (Intellectica 51 (2009), 169-189).

I can’t have claimed that I have established that CH is neither a definite mathematical problem nor a definite logical problem, since one can’t say precisely what such problems are in either case.  Rather, as workers in mathematics and logic, we generally know one when we see one.  So, the Goldbach conjecture and the Riemann Hypothesis (not “Reimann” as has appeared elsewhere in this exchange) are definite mathematical problems.  And the decidability of the first order theory of the reals with exponentiation is a definite logical problem.  (Logical problems make use of the concept of formal language and are relative to models or axioms.) Even though CH has the appearance of a definite mathematical problem, it has ceased to be one for all intents and purposes because it was long recognized that only logical considerations could be brought to bear to settle it, if at all.  So then what would make it a definite logical problem? Something as definite as: CH is true in L.  I can’t exclude that some time in the future, some model or axiom system will be produced that will be as canonical in nature for some concept of set as L is for the concept of hereditarily predicatively definable set.  But I’m not holding my breath either.

I don’t know whether your concept of set-theoretical truth can be assimilated to Maddy’s A-realism, but in either case I see it as trying to have your platonist cake without eating it.  It allows you to accept CH v not-CH, but so what?

Best,
Sol

# Re: Paper and slides on indefiniteness of CH

Dear Penny,

On Tue, 5 Aug 2014, Penelope Maddy wrote:

Thank you for the plug, Sol.  Sy says some interesting things in his BSL paper about ‘true in V':  it doesn’t ‘reflect an ontological state of affairs concerning the universe of all sets as a reality to which existence can be ascribed independently of set-theoretic practice’, but rather ‘a façon de parler that only conveys information about set-theorists’ epistemic attitudes, as a description of the status that certain statements have or are expected to have in set-theorist’s eyes’ (p. 80). There is ‘no “external” constraint … to which one must be faithful’, only ‘justifiable procedures’ (p. 80); V is ‘a product of our own, progressively developing along with the advances of set theory’ (p. 93).  This sounds more or less congenial to my Arealist (a non-platonist):   in the course of doing set theory, when we adopt an axiom or prove a theorem from axioms we accept, we say it’s ‘true in V’, and the Arealist will say this along with the realist; the philosophical debate is about what we say when we’re describing set-theoretic activity itself, and here the Arealist denies (and the realist asserts) that it’s out to discover the truth about some objectively existing abstracta.  (By the way, I don’t think ‘truth-value realism’ is the way to go here.  In its usual form, it avoids abstract entities, but there remains an external fact-of-the-matter quite independent of the practice to which we’re supposed to be faithful.)

My apologies here. In my reply to Sol I only made reference to truth-value realism for the purpose of illustrating that one can ascribe meaning to set-theoretic truth without being a platonist. Indeed my view of truth is very far from the truth-value realist, it is entirely epistemic in nature.

Unfortunately the rest of my story of the Arealist as it stands won’t be much help because the non-platonistic grounds given there in favor of embracing various set-theoretic methods or principles are fundamentally extrinsic and Sy is out to find a new kind of intrinsic support.

Yes. I am trying to make the case that there are unexplored intrinsic sources of evidence in set theory. Some have argued that we must rely solely on extrinsic sources, evidence emanating directly from current set-theoretic practice, because intrinsic evidence cannot take us past what is derivable from the maximal iterative conception. I do agree that this conception can lead us no further than reflection principles compatible with V = L.

But in fact my intuition goes further and suggests that no intrinsic first-order property of the universe of sets will enable us to resolve problems like CH. We have to examine features of the universe of sets that are only revealed by comparing it to other possible universes (goodbye Platonism) and infer first-order properties from these “higher-order” intrinsic features of V (a name for the epistemically-conceived universe of sets).

Obviously a direct comparison of V with other universes is not possible (V contains all sets) so we must instead content ourselves with the comparison of pictures of V. These pictures are perfectly provided by the hyperuniverse (also conceived of non-platonistically). And by Löwenheim-Skolem we lose none of the first-order features of V when we model it within the hyperuniverse.

Now consider the effect that this has on the principle of maximality. Whereas the maximal iterative concept allows us to talk about generating sets inside V by iterating powerset “as long as possible”, the hyperuniverse allows us to express the maximality of (a picture of) V in a more powerful way: maximal means “as large as possible in comparison to other universes” and the hyperuniverse gives a precise meaning to this by providing those “other universes”. Maximality is no longer just an internal matter regarding the existence of sets within V, but is also an external matter regarding the largeness of the universe of sets as a whole in comparison to other universes. Thus the move from the concept of set to the concept of set-theoretic universe.

Now comes a crucial point. I assert that maximality is an intrinsic feature of the universe of sets. Certainly I can assert that there is a rich discussion of maximality in the philosophy of set theory literature with some strong advocates of the principle, including Goedel, Scott and yourself (correct me if I am wrong).

Maximality is not the only philosophical principle regarding the set-theoretic universe that drives the HP but surely it is currently the most important one. Another is omniscience (the definability in V of truth across universes external to V). Maybe there will be more.

I’m probably insufficiently attentive, or just plain dim, but I confess to being confused about how this new intrinsic evidence is intended to work.   It isn’t a matter of being part of the concept of set, nor is it given by the clear light of mathematical intuition.  It does involve, quoting from Gödel, ‘a more profound understanding of basic concepts underlying logic and mathematics’, and in particular, in Sy’s words, ‘a logical-mathematical analysis of the hyperuniverse’ (p. 79).  Is it just a matter of switching from the concept of set to the concept of the hyperuniverse?  (My guess is no.)  Our examination of the hyperuniverse is supposed to ‘evoke’ (p. 79) certain general principles (the principles are ‘based on’ general features of the hyperuniverse (p. 87)), which will in turn ‘suggest’ (pp. 79, 87) criteria for singling out the preferred universes — and the items ultimately supported by these considerations are the first-order statements true in all preferred universes. One such general principle is maximality, but I’d like to understand better how it arises intrinsically out of our contemplation of the hyperuniverse (at the top of p. 88).  On p. 93, the principle (or its more specific versions) is said to be ‘the rigorous expression of what it means for an element of the hyperuniverse, i.e., a countable transitive model of ZFC, to display “maximal properties”‘.  Does this mean that maximality for the hyperuniverse derives from a prior principle of maximality inherent in the concept of set?

You ask poignant questions; I hope that what I say above is persuasive!

Many thanks for your interest, and very best wishes,
Sy

# Re: Paper and slides on indefiniteness of CH

Dear Sol,

On Sun, 3 Aug 2014, Solomon Feferman wrote:

Dear Sy,

Thanks for your helpful comments on my draft, “The Continuum Hypothesis is neither a definite mathematical problem nor a definite logical problem,” and especially for bringing your Hyperuniverse Program (HP) to my attention.  I had seen your 2013 article with Arrigoni on HP back then but had not taken in its point.  I have now read it as well as your Chiemsee slides, and will certainly take it into account in the final version of my paper. I’m glad that we are in considerable agreement about my fundamental argument that one must distinguish mathematical problems in the ordinary sense from logical problems, and that as of now what I claim in the title is true, even taking HP into consideration.  Is my title misleading since it does not say “as of the time of writing”? The reader will see right away in the abstract and the opening section that what I claim does not exclude the possibility that in the future CH will return as a definite mathematical problem [quite unlikely] or that it will somehow become a definite logical problem.

This does appear to constitute a significant retreat in your position. In the quote of yours that I used in my Chiemsee tutorial you refer to CH as being “inherently vague”, in other words dealing with concepts that render it impossible to ever assign it a truth value. If you now concede the possibility that new ideas such as hinted at by Gödel in the quote below (and perhaps provided by the hyperuniverse programme) may indeed lead to a solution, then the “inherent vagueness” argument disappears and our positions are quite close. Indeed we may only differ in the degree of optimism we have about the chances of resolving ZFC-undecidable problems in abstract set theory through philosophically-justifiable logical methods.

“(Gödel) Probably there exist other axioms based on hitherto unknown principles … which a more profound understanding of the concepts underlying logic and mathematics would enable us to recognize as implied by these concepts.”

This is not the place to respond to your many interesting comments on the draft, nor on the substance of the HP and your subsequent exchange with Woodin.  But I would like to make some suggestions regarding your terminology for HP (friendly to my mind).  First all, it seems to me that “preferred models” is too weak to express what you are after.  How about, “premier models” or some such?  (Tapping into the Thesaurus could lead to the best choice.)

I do see your point here, because I do want to suggest not simply a “preference” for certain universes over others but rather a “compelling” or “justified” preference. I’ll give the terminology more thought, thanks for the comment.

Secondly, I’m not happy about your use of “intrinsic evidence for set-theoretic truth” both because “intrinsic evidence” is commonly used to refer to the constellation of Gödel’s ideas in that respect (not the line you are taking) as opposed to “extrinsic evidence”, and because “set-theoretic truth” suggests a platonistic view (which you explicitly reject).  I don’t have anything to take its place, but it reminds me of the kinds of methodological maxims that Maddy has promoted, so perhaps a better choice of terminology can be found in her writings in place of that.

I do not think that “set-theoretic truth” entails a platonistic viewpoint (indeed there is a concept of “truth-value determinism” that falls short of Platonism). The goal of the programme is indeed to make progress in our understanding of truth in set theory and a key claim is that there is intrinsic evidence regarding the nature of the set-theoretic universe that transcends the older form of such evidence emanating from the maximal iterative conception. I think that the dichotomy intrinsic (a priori) versus extrinsic (a posteriori) which Peter Koellner has emphasized is a valuable way to clarify the debate. Nevertheless I do appreciate that some have suggested that the distinction is not as sharp as I may have assumed and I would like to hear more about that.

Another very interesting question concerns the relationship between truth and practice. It is perfectly possible to develop the mathematics of set theory without consideration of set-theoretic truth. Indeed Saharon has suggested that ZFC exhausts what we can say regarding truth but of course that does not force him to work just in ZFC. Conversely, the HP makes it clear that one can investigate truth in set theory quite independently from set-theoretic practice; indeed the IMH arose from such an investigation and some would argue that it conflicts with set-theoretic practice (as it denies the existence of inaccessibles). So what is the relationship between truth and practice? If there are compelling arguments that the continuum is large and measurable cardinals exist only in inner models but not in V will this or should this have an effect on the development of set theory? Conversely, should the very same compelling arguments be rejected because their consequences appear to be in conflict with current set-theoretic practice?

Best wishes and many thanks,
Sy