# Can the continuum hypothesis be settled in physics?

Can the continuum hypothesis be settled in physics? In a lecture mathematician Woodin considers the possibility:

Develops the mathematical physics of a mathematical understanding of the physical universe. If it starts to need large Cardinals remember large carnal axioms with finite-istic consequences so it's not completely unreasonable that large Cardinal axioms provide mathematical truths that you need to do the analysis of physics we already saw an instance of that infinitely many wooden Cardinals imply that the projective continuum hypothesis is true that's a remarkable connection between very large sets and very small sets who's to say that doesn't happen in physics somewhere so that would be a win and I'll tell you I would be as stunned.

What are philosophers take on such kind of claims? (feel free to include references) Are they genuinely considered?

• Comments are not for extended discussion; this conversation has been moved to chat. Jan 30, 2023 at 9:15
• Cough cough ... Can an answer be updated? ;p Mar 20 at 10:38
• An answer can be edited and updated. This is unrelated to comments being moved to Chat. Mar 21 at 8:59

In physics you make observations. You then try to write down equations that describe these observations. The simpler the equations the better. These equations are called a model.

Physics can only settle an issue by providing a model that does not break when we plug in extra data. The closest physics can come to settling the continuum hypothesis is to create a mathematical model that requires we assume the existence of a set of a given infinite size, and that model is somehow better than those without the assumption.

For an easier example consider how most models contain the set of real numbers ℝ in some form. Space is assumed to be continuous. It is a product of real lines. This allows us to integrate the forces on a particle over a given path and get the change in energy. The change in energy and the forces are all made of real numbers.

These theories work. Does that settle the issue of whether the real numbers exist? Or does it merely imply they are a good mathematical convenience?

Perhaps there is a better model where ℝ is replaced with some exotic type of ordered set with more elements than the natural numbers but fewer than ℝ itself. Would such a model prove the continuum hypothesis?

Even the existence of infinity is tenuous in physics. The infinite sets that occur there are all sets of infinitely many possibilities. There are continuum-many values the energy of a particle could take. But we have only ever measured finitely many of those values.

From a mathematical point of view, it would be unlikely to find a box containing uncountably many particles. Countably many perhaps. But if we add up the contribution of uncountably many things it ends up that all but countably many are zero.

The main philosophical positions here are the various versions of mathematical platonism and...those who aren't platonist, like logicism and formalism. But really the relevant distinction here is the distinction between realism and non-realism (or anti-realism):

Do we believe that mathematical abstract objects (such as sets) have an existence independent of the human mind?

Platonists - and realists in general - say "yes", but they differ greatly in what exactly such "existence" means. For example in re structuralists would probably find the suggestion that mathematical statements about sets can be investigated by looking at the physical world rather plausible, since they have to believe that there are physical structures exemplifying all mathematical constructs.

Formalists, logicists and other non-realists would find the suggestion absurd: To them, mathematical objects and statements have no existence independent of the formulae used to describe them; the "truth" of a mathematical statement is not a statement about anything that exists at all, but statements about what can be abstractly derived given a particular axiom set and a particular set of rules of deduction.

Concerning set theory generally and statements about infinite sets like the continuum hypothesis specifically, realists run into a problem similar to the problem of induction in the philosophy of science: How does one, in practice, exhibit an infinite set or a map between infinite sets? The structuralist who believes there must be physical structures exemplifying an infinite set can, by the nature of infinity, never even finish recounting the members of even one of these sets!

Not all realists despair at this problem, but a notable fraction become (ultra)finitists who do believe in the existence of mathematical objects but not in infinite objects.

• I think the question is about mathematical structure of theories in physics and not about concrete objects in the nature. As an example, axion of choice is equivalent to the existence of a basis for infinite dimensional Hilbert spaces, so one can argue that infinite dimensional Hilbert spaces are necessary for quantum physics and as a result, physics can provide some evidence for truth of axiom of choice. There may be some similar story for continuum hypothesis. Jan 29, 2023 at 18:38
• @Arian 1. The axiom of choice and the existence of infinite-dimensional orthonormal bases are not known to be equivalent - AC implies the existence of the basis, but the converse is not known, cf. this math.SE question. 2. You are implicitly taking a realist view here if you think that the phrase "evidence for truth of axiom of choice" actually means something. To a non-realist, this just doesn't make any sense - the axiom of choice is not a claim about reality, so it cannot have "evidence" for or against it. Jan 29, 2023 at 19:39
• (The idea that because the physical theory that models reality requires the axiom of choice this constitutes "evidence" for the axiom of choice is close to a in re structuralist view that reality then "exemplifies" the axiom of choice) Jan 29, 2023 at 19:40
• Thanks, I mean Blass theorem and I mistakenly referred to it. You should take truth as correspondence and then say I am a realist. I also remembered that there are some papers on continuum hypothesis and some model of hidden variable quantum mechanics, arxiv.org/abs/1212.0110. Jan 29, 2023 at 20:12
• @Arian how important are bases for quantum mechanics? Jan 29, 2023 at 21:38

The continuum hypothesis is already settled. The answer to it is that there are models of set theory in which the continuum hypothesis is true, and there are also models of set theory in which the continuum hypothesis is false. Both kinds of models are fully valid and self-consistent; there is nothing wrong with either kind of model.

By analogy, is up positive and down negative, or is down positive and up negative? Both systems are fully valid and self-consistent; there is nothing wrong with either one.

That said, there's still a meaningful question here: does physics provide us with some reason to favor one kind of model of set theory over the other? Is one kind better or more useful than the other one when we're talking about physics?

So far, the answer seems to be no. As far as I know, there is no obvious connection between the mathematics describing the laws of physics, and the question of how many cardinalities there are among infinite sets of real numbers. If one mathematician says "yes" to the continuum hypothesis and another one says "no," then neither mathematician will have an easier or harder time with physics as a result of that choice.

Perhaps in the future, some new laws of physics will be discovered that are related to cardinalities of infinite sets of real numbers, and we will find that these laws of physics are particularly easy to describe if we take the continuum hypothesis to be true, but hard to describe if we take the continuum hypothesis to be false—or the opposite. (My gut feeling says that that's pretty unlikely.)

Even if the laws of physics turn out to favor models of set theory where the continuum hypothesis is true (or false), that won't mean that in our universe, the continuum hypothesis is true (or false). One of the two options may turn out to be less useful for physics, but being less useful doesn't mean that it's wrong.

• Indeed CH holds in some models and not others. I interpret the question to be about whether or not CH holds in the universe of all physical objects. Jan 31, 2023 at 1:40

One underdeterminate epistemic possibility in this connection is that a theory about perception itself might involve making a judgment about the cardinality of the set of real numbers. The open coloring question, for example, might conceivably be relevant to a theory of color perception (albeit quite in abstracto), and since two samples of open-coloring axioms in tandem resolve the natural powerset to ℵ2, one might reason from an attendant theory of color perception (if it's actually possible/relevant) to the conclusion that the natural powerset is ℵ2.

Another (not-so-clear) possibility might be that forcing as a mathematical phenomenon has a physical counterpart, and this counterpart can change the size of physical continua. I've been working on an attempt to model laws-of-physics on infinite conjunctions in infinitary logic where for t (time) = n, our physical universe has an infinitary logical signature ℒ(ωf(n), ωg(n)) such that some functions f and g yield an evolution of the physical world's logical signature over time (this is to try to implement Lee Smolin's changing-laws-of-physics idea), but to be honest, I haven't gotten much further than a nifty background for a science fiction storyline, not a genuine scientific hypothesis. (I.e., what predictive value does this "model" have, if any? The best I've thought of would be that our capacity for continuous perception would change with the changing cardinality of physical continua, but how would that be "noticeable" or evaluable, then?)

Broadly, one problem with thinking that mathematical physics, much less experimental physics that's mathematically informed, would be amenable to novel reasoning for or against CH is that there are an extremely vast number of alternatives to CH in higher set theory. ZFCwise, there are absolutely infinitely many alephs that the natural powerset can be forced to equal, and beyond the edge of ZFC, there is even the possibility of forcing the Continuum to equal absolute infinity itself (see also Timothy James' essay on predicativism in the philosophy of mathematics on the "indefinite extensibility" of the natural powerset; or consider that the surreal number line itself contains absolutely infinitely many infinitesimals in every interval, incl. [0, 1]). Perhaps physics would at least allow us to eliminate the prior disjunct in {CH ∨ ~CH} but the latter disjunct is so internally vast that said elimination would be as meager a contribution to the issue as possible. Even worse, it's not only that the natural powerset can be forced to equal so many things on its own, but: the powerset of the zeroth aleph can be forced to equal the powerset of the first aleph, as well as the second, third, fourth, etc. alephs, and so indeed, modulo the proper-class scales of options, we might force every well-ordered transfinite cardinal, prior to ℶ1, to equal ℶ1, so that the first beth is a fixed point of the aleph function. On the surface, it is hard to say how empirical information, or mathematical models of said information, would include strong, clear reasons for filtering in, or out, so many options.

Even more insidiously, suppose that the well-ordering principle is waived (because the basic axiom of choice, or whichever choice axiom, is waived). Then it is possible for the cardinality of the Continuum to be a transfinite cardinal, but not from the well-ordered sequence of such cardinals, i.e. it would not be an aleph but perhaps similar to (apparently not identical to, though) an amorphous set.

Perhaps most insidiously of all, suppose you waive the powerset axiom itself. You can still use the classical diagonal argument to show that the set of real numbers is not bijective with the set of natural numbers, but you no longer have that 20 is the arithmetical expression of this difference. (I don't think this is plausible at all, since I think John Conway's explanation for the exponential expression fitting the case is a perfectly apparent explanation, but on the other hand, see again James' predicativist apologetics for how the exponential function on whichever X can come apart from the concept of "the set of all subsets of X.")

EDIT: Though this answer was accepted and has received a number of upvotes, it was also downvoted, and I feel like I worded it in an imprecise way. Firstly, on the "yes" side, my references to open coloring axioms and physical forcing are very speculative; insofar as reductionism is not in vogue anymore, I imagine that biological/neurological theories of color perception might indeed be relevant to physical theories in a way that could also play into evaluating some version of the Continuum problem, but I am not especially well-versed in actual physics, biology, or neurology, so I feel like I should emphasize just how speculative my comments on this score are.

Second, on the "maybe not" side: overall, I do not think that there is really just one powerset function. Cantor's theorem has it that various sets of subsets must exceed their bases, but complications involving definable vs. hyperdefinable (or even antidefinable) subsets seem to multiply the question of such a function. I think that the enduring desire to "settle" the Continuum issue is often derived from the seeming continuity involved in physical perception, so that, "How many points are there in physical space?" seems like a question of external/objective reality. So rather than say that every powerset function might be resolved by some future theory of physics, I would rather say that something like "the set of all physically realizable subsets of a countable set" would be the specific subtype of the powerset function whose size could be "settled" in such a theory. But this too is speculative, after all; I offer these conjectures as an answer to the OP question in only the bare sense that they address the modal term in that question per the title of the OP post: i.e., in some abstract sense of "can," CH, or a version of it anyway, "can" be settled by physics.

Finally, there is one more "insidious" variation on the "maybe not" theme that comes to my mind, one based on paraconsistent set theory. In PST, perhaps, one might force the Continuum's cardinality to equal several inconsistent numbers at once; applying this "model" to physics, or physics to this "model," would then mean bringing in considerations like a paraconsistent theory of superposition. Paraconsistent logic is motivated by the desire to avoid an inferential explosion, and forcing the Continuum's cardinality to equal every option otherwise delineated above is almost the same as (or maybe even identical to, eventually) such an explosion; so a paraconsistent set theorist would still be motivated to include some restrictions on their "insidious" standpoint, to rule out the most deviant alternative to CH of all.

But so again, overall, for a theory of physics to settle any version of the Continuum question would mean that such a theory would have to sort through questions about choice axioms, forcing, the geometry of perception, etc., all in a way that comports with how theories of physics are reliably set up. I'm not a physicist and I am reluctant to press my claims, here, too strongly, out of concern for veering off into pseudoscientific territory. I appreciate that my answer was accepted, and I think my answer involves informed reflection on the parameters of the OP question, but insofar as all this is philosophical reflection, I am still highly uncertain about my conclusions in this case.

• @user4894 off the top of my head, I don't know of any "obviously philosophical" essays about positively using physics theories to address CH. However, against using such theories in that way, well... that's where the greater weight of my post lies. Predicativism, for example, is a school of philosophy of math that brings into sharp relief some of the issues with portraying CH as an objective matter, and if CH is not so objective, it will perhaps be harder to link its settlement to more objectively-minded theories of physics. Jan 29, 2023 at 22:50
• as you are, per your edit, concerned with the downvotes, here's my reason: while I cannot judge the quality of your question, I can confidently say that unless a reader is already deeply familiar with the very specific topics you bring up, he won't understand much of it, nor be able to bring it in contact with the question. The other two answers show how a less deep level of detail can answer the question just as well, with the added benefit of average readers (who are not deep into mathematics of infinity) being able to understand them relatively easily.
– AnoE
Jan 30, 2023 at 13:17
• @AnoE fair enough, except maybe not-upvoting would be more fair than actively-downvoting. I know this is the PhilosophySE, not the PhilosophyOverflow (and there isn't a PhilosophyOverflow), but so I still provided links for pretty much every technical detail I brought up, and I don't see why answers with fewer citations would really be better fits for the SE network as whole than one which pools together a great deal of relevant information. Jan 30, 2023 at 20:40
• @KristianBerry i am considering removing your answer from the accepted answer because of it might be bringing unwanted attention. I also suspect many people downvote when they do not agree (but in that case i would recommend commenting or answering while mentioning). If ur okay with the downvotes onslaught I'll let it remain? Jan 31, 2023 at 4:28
• @MoreAnonymous yeah you can unaccept my answer, if I can think of improvements to my answer I will make them but for the time being I don't think my answer is helpful. Perhaps I will just delete the answer altogether. Jan 31, 2023 at 8:40