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The present post is intended to tackle the possible interactions of two bizarre realms of extremely large and extremely small creatures, namely large cardinals and quantum physics.

Maybe after all those paradoxes and uncertainty phenomena among weird tiny particles, which follow their own weird quantum logic, and after all those controversies surrounding the right interpretation of what is going on in the sub-atomic universe, the last straw that would break the camel's back could be the discovery of a series of statements in quantum theory which are independent or have large cardinal strength set theoretically. The fact that will send such physical statements beyond the realm in which the so-called usual mathematical tools can afford us a solution.

Not to mention that inspired by Hilbert's sixth problem and Godel's incompleteness theorems, some prominent physicists already brought up discussions concerning the possibility of obtaining independence results or existence of undecidable facts/theories in physics. In this direction see Stephen Hawking's lecture, Godel and the end of universe. [The corresponding post on MSE might be of some interest as well].

Anyway the bad (good?) news is that the intersection of large cardinal theory and quantum physics is non-empty (if not potentially large). For example one may consider the following theorem of Farah and Magidor in the Independence of the existence of Pitowsky spin models which contains an assumption of consistency strength of measurable cardinals. [cf. R. Solovay, Real-valued measurable cardinals, Axiomatic Set Theory, 1971.]

Theorem (Farah - Magidor): If the continuum is real-valued measurable then Pitowsky's kind spin function does not exist. The same holds in the model one gets from any universe of $ZFC$ by adding $(2^{\aleph_0})^+$-many random reals.

See also some related philosophical discussions regarding this result in:

One also might be interested in taking a look at the following papers which shed some light on the way forcing, Cohen reals, ultrafilters and various set theoretic concepts and tools play role in connection with some problems of quantum physics including hidden variables:

Inspired by Farah-Magidor's theorem and the other mentioned papers, the following question arises:

Question: What are some other examples of the statements in (quantum) physics which are mathematically independent or have some large cardinal strength (or at least make use of large cardinal assumptions in their formulation)?

Please provide references if you are aware of any such result.

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    $\begingroup$ EPR is not a paradox. It is just the way the world is. $\endgroup$ – Jeff Harvey Aug 29 '17 at 23:57
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    $\begingroup$ @JeffHarvey Please note that the existence of physical statements which are independent or have large cardinal strength DOESN'T mean that there are some paradoxes in physics necessarily. They actually don't imply such a matter in mathematics as well. In fact such statements simply indicate that some necessary axioms for understanding the universe are missing or there are some fundamental barriers to our knowledge of the universe based on the current framework/model/paradigm. It is similar to Heisenberg's uncertainty barrier which prevents us from getting the full information as we like. $\endgroup$ – Morteza Azad Aug 30 '17 at 8:01
  • $\begingroup$ @Heinrich Sorry, but I can't figure out what you mean properly! Could you please add more explanations? What do you mean by non-existence of full information? Are you pointing out to a physical phenomenon similar to black hole information loss? $\endgroup$ – Morteza Azad Aug 30 '17 at 22:49
  • $\begingroup$ @Heinrich I am somehow familiar with the views in quantum mechanics suggesting that reality doesn't exist until we measure it. It resembles the set theoretic multiverse view which suggests something of the same spirit about nature of mathematical reality that in this view is undeterministic and depends on the mathematical model that you live in. Find more along these lines in the lectures of the Set Theoretic Pluralism symposiums. $\endgroup$ – Morteza Azad Aug 31 '17 at 15:51
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In a recent result with Shay Moran, Pavel Hrubes, Amir Shpilka and Amir Yehudayoff, we show that the answer to a basic question in statistical machine learning is determined by the value of the continuum, and is therefore independent of the ZFC set theory. The paper will be available on Arxiv within a few weeks. Here is a link to that paper: https://arxiv.org/abs/1711.05195

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    $\begingroup$ (+1) Welcome to MathOverflow, Shai! That is really interesting! We look forward to reading the full text as soon as it appears on Arxiv. Please don't forget adding a link here as well. $\endgroup$ – Morteza Azad Aug 28 '17 at 21:31
  • $\begingroup$ @Heinrich Here is a related post on MathOverflow concerning the actual size of continuum. The thread summarizes some other related posts as well. $\endgroup$ – Morteza Azad Aug 30 '17 at 22:55
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Sorry, not an answer, but too long for a comment.

I am the author of the "Pitowsky's Kolmogorovian models and Super-Determinism" paper. I would reject the claim that this paper is a "philosophical discussion". Rather, I (try to) demonstrate (using simple physics and mathematics, not philosophy) that the whole Pitowsky model business is physically meaningless (and accordingly not a good example for a connection between set theory and physics).

Let me give a comparison: One can prove that no strategy will guarantee a steady income playing roulette, but of course for such a proof you have to assume your potential strategy is "measurable". One can now claim "Aha! This indicates a deep connection between financial mathematics and set theory!" Which of course would be silly.

The Pitowsky construction is more complicated, and therefore it is harder to see, but basically he does something similar: He sabotages the simple, elegant proof of Bell's theorem by assuming that some stuff is not measurable. Of course such an assumption does not constitute a hidden variable theory (just as claiming "the winning strategy might not be measurable" only sabotages the roulette-proof, but doesn't give you a winning strategy). Pitowsky then goes on to actually "construct" such non-measurable hidden variables, by introducing a non-standard notion of probability. But, as I try to point out in my paper, in the end this notion just says: "We assign (nonstandard) probability 1 to exactly those results that will come out of those of experiments that will actually be performed". This is logically consistent, but basically equivalent to super-determinism: Once we know exactly what will happen (in particular: which measurement will be performed), there is no locality problem with Bell or GHZ. This has been obvious from the the beginning of the investigations of no-go theorems.

To come back to the comparison: I can just as well claim to have a roulette winning strategy: I will always play the color that will then be picked. Again, this doesn't seem to me to indicate a connection between financial mathematics and the axiom of choice...

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  • $\begingroup$ (+1) Thanks for sharing your point of view and contributing to this thread, Jakob! Well, philosophically speaking I didn't claim that your technical paper is of philosophical nature entirely. However some of its aspects are borderline philosophical in the sense of being related to the general question about the interaction of mathematics and physics in philosophy of mathematics and science. $\endgroup$ – Morteza Azad Sep 14 '17 at 11:22
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It was shown just two years ago that the presence or absence of a spectral gap of certain short-ranged 2D lattice Hamiltonians is independent of the ZFC axioms:

T. S. Cubitt, D. Perez-Garcia, and M. M. Wolf, "Undecidability of the Spectral Gap", Nature 528, 207-211 (2015)

(A 146-page-long full version can be downloaded from this link.)

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    $\begingroup$ Worth mentioning that the result in the paper actually implies that for any (recursively axiomatizable) theory some instance of this problem is undecidable. Nothing about ZFC or large cardinals is at play here. $\endgroup$ – Wojowu Aug 29 '17 at 21:06
  • $\begingroup$ @Wojowu: you are completely right. Thanks for this comment. $\endgroup$ – Zoltan Zimboras Aug 29 '17 at 21:11
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    $\begingroup$ Yeah, I'm pretty sure they just found a way to encode certain mathematical statements in spectral gaps. Its similar to running a computer testing Con(ZFC). $\endgroup$ – PyRulez Nov 17 '17 at 18:58
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Though this does not directly answer your question, here is a foundational paper that might help one derive results that might answer your question:

Marian Boykan Pour-El and Ian Richards: "Noncomputability in Analysis and Physics: A Complete Determination of the Class of Noncomputable Linear Operators", Advances in Mathematics 48, 44-74 (1983).

I quote the short first paragraph of this paper as it sets the tone for what follows:

"One would assume that a "reasonable" operator should map computable input data onto computable solutions. It is perhaps surprising that many of the standard operators of analysis and physics fail to do this. In this article, we shall determine precisely which linear operators do, and which do not, preserve computability."

I hope this paper helps.

Addendum: Consider their Main Theorem and its Complement:

Main Theorem: Let $X$ and $Y$ be Banach spaces with computability theories, and let $e_{n}$ be an effective generating set for $X$. Let $T$: $X$$\rightarrow$$Y$ be a closed linear operator whose domain includes {$e_{n}$} and such that $T$$e_{n}$ is a computable sequence in $Y$. Then $T$ maps every computable element of its domain onto a computable element of $Y$ if and only if $T$ is bounded.

Complement. Under the same assumptions, if $T$ is bounded then more can be said. The domain of $T$ coincides with $X$, and $T$ maps every computable sequence in $X$ onto a computable sequence in $Y$.

Working backward to discover the philosophical motivation for their first paragraph, it seems the place to start in order to analyze the non-computabilty in analysis and physics they seek to show.

Does this help any, Morteza?

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  • $\begingroup$ (+1) It is related and interesting! Thanks for sharing! It makes me think how reasonable the premise in this quotation is actually. I mean the following part: "One would assume that a "reasonable" operator should map computable input data onto computable solutions." I am not sure what this philosophical judgement is based on exactly. Do you have any idea, Thomas? $\endgroup$ – Morteza Azad Aug 31 '17 at 10:13
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Not sure if it fits what you're asking, but the QM prediction that a physical apparatus can generate a stream of random bits is independent, or anyway scientifically unverifiable, since Kolmogorov complexity is uncomputable. If you flip a coin 1000 times and assert that the resulting string has complexity > 900 bits, that is almost certainly true, but in that case it's unprovable (example based on a similar observation by Leonid Levin). Not only that, the stream is supposed to be arithmetically random and not just Kolmogorov ($\Pi^0_1$) random, i.e. you can't compress it even with a $0^{(n)}$ oracle for finite $n$. I don't know what happens if $n\ge \omega$.

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I don't know how relevant this is to your question, but Stanisław Ulam had some speculations on physical applications of set theory in

S. Ulam, Combinatorial analysis in infinite sets and some physical theories, SIAM Review, Vol. 6, No. 4 (Oct., 1964), pp. 343–355.

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  • $\begingroup$ (+1) So exciting! While I have been well-aware of Ulam's works in set theory and physics (particularly the development of Teller–Ulam design of thermonuclear weapons), I didn't know anything about Ulam's personal view towards possible applications of set theory in physics. Though, in the view of military projects that Ulam had been involved in, one may speculate on the peaceful nature of the applications that he had in mind.Who knows? Maybe he was desiring to design an $\omega$-megaton nuclear weapon, an Omega-Bomb! :-| $\endgroup$ – Morteza Azad Aug 31 '17 at 11:40
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The examples in this thread are interesting as curiosities, but while one never knows where fundamental explorations will lead, we should be skeptical of their physical applicability.

In physics, the same observable theory may have very different formulations. For example, we may have a formulation using a continuum, and a formulation using a limit of discrete approximations. Given the finite and approximate nature of observations, we should be skeptical if the discrete approximations do not converge to the continuum limit (or if singularities lead to nonrecursive computations). It would be remarkable if physical distance π is observably different from π-ε.

Now, consider two physical theories, where theory A depends on basic arithmetic and theory B depends on the Continuum Hypothesis (CH), with both A and B giving the same predictions (assuming CH). Even if the theory is experimentally confirmed, without more, we cannot say that the Continuum Hypothesis was resolved. What would have more value is a series of equivalences that point to a coherent vision of set theory, or even better, the ability to run experiments to test arbitrary set theoretical propositions of a certain type.

As for possible examples, mathematical consistency of a nontrivial relativistic 4-dimensional quantum field theory is an open problem. Thus, for all we know, it might be equiconsistent with a supercompact cardinal, though there is no present evidence to that effect. We also cannot yet rule out that the theory is consistent but only for nonrecursive configurations.

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  • $\begingroup$ Thanks for your feedback, Dmyrto! While the idea of searching for a resolution of the Continuum Hypothesis (and other undecidable set theoretic assumptions) through physical observation is what Magidor has already suggested in his "Some set theories are more equal" paper (linked in my post), but this is not exactly what I am asking in this post. Here I simply look for bridges that (even potentially) can connect the two seemingly separated realms of large cardinal and quantum theory; bridges like real numbers and $C^*$-algebras. The former could be an application of the latter, though. $\endgroup$ – Morteza Azad Aug 31 '17 at 18:40
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In my opinion one of the best examples of a physical statement depending on a mathematically independent statement is provided by Malament-Hogarth machines (You can find a bunch of other links just by doing a google scholar search). These are machines which it is argued (though others argue they are impossible) are allowed by general relativity and allow the operator to determine the answer to any $\Pi^0_1$ claim in finite time. As, by Godel's theorem, given any computably axiomitizeable theory $T$ the $\Pi^0_1$ statement $Con(T)$ is independent from $T$. So whatever theory one chooses to work in this lets you produce a physical statement (the machine with such and such construction ... will give answer blah) that depends on the truth of an independent question.

Note that this doesn't specifically involve large cardinal assumptions themselves but it does give you a physical system whose outcome depends on the consistency of large cardinal statements (e.g. $Con(ZFC+exists measurable)$)

However, one should not assume that how these physical systems turn out tells us what is mathematically true. We develop physical theories by accepting those hypothesises that seem to be good descriptions of reality and it is usually convenient to write those theories in terms of the most natural mathematical structures like the natural numbers or the reals.

However, it is equally true that the theory which says experiments turn out the way General Relativity formulated in this non-standard model of the reals/integers predicts (almost certainly it can be formalized in the two-sorted first-order theory of second order arithmetic but if not use a non-standard model of ZFC instead) is as compatible with all our evidence as the theory which applies General Relativity formulated in the standard model.

Ohh to put it more simply if your Malament-Hogarth machine tells you that a particular computation converges in finite time you can't really be sure that the computation really converges in finite time or if the temporal structure of the universe is non-standard and the computation only converges at some non-standard time.

Thought for a perspective which argues that we could use the results of these MH machines to determine mathematical truth see Sharon Berry's Malament–Hogarth Machines and Tait's Axiomatic Conception of Mathematics with preprint here. (Conflict of interest warning: I'm married to Sharon though I disagree with her conclusions).

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  • $\begingroup$ (+1) Thanks for your comprehensive answer, Peter! Just as a possibly irrelevant question inspired by your comment about general relativity, I wonder whether we can consider the singularity phenomena in the universe such as what is going on at the center of black holes or what has happened right at the very beginning of the universe (the big bang moment), as physical situations which are independent of the known principles of physics? If yes, can we describe such phenomena via independent statements in mathematics? $\endgroup$ – Morteza Azad Nov 23 '17 at 0:14
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    $\begingroup$ Not really ... those are kinda different issues. The only way it makes sense to use an independent mathematical statement to represent some physical situation is when we have good reason to believe that we understand the way in which physical laws apply but our mathematical axioms just aren't good enough to let use determine the outcome e.g., we can predict experimental outcome A if con(PA) and B if not. Also, we have to believe there really is a mathematical fact of the matter (the statement there are at least 3 elements is independent of axioms of group theory but its not true or F $\endgroup$ – Peter Gerdes Dec 19 '17 at 9:39
  • $\begingroup$ The situation you are talking about seems to be one in which we don't believe our physical theories accurately capture reality. Not one in which they do capture reality but our axioms just won't let us work out what the answer is. Just like in algebra the fact that I don't know what x is and I don't know what y is doesn't mean x and y are the same. $\endgroup$ – Peter Gerdes Dec 19 '17 at 9:41
  • $\begingroup$ I suppose for circumstances where we know the physical reality has to choosen from 2^n possibilities you could choose n mutually independent statements over ZFC and simply stipulate that however the physics turns out you will adopt the axioms which give that answer. But that doesn't do what you want...it just re-describes the choice between possible physical theories as an arbitrary, unrelated and uninformative choice about which model of set theory we want to work in. $\endgroup$ – Peter Gerdes Dec 19 '17 at 9:43

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