# Applications of set theory in physics

In the introduction of the paper "Links between physics and set theory", the following quote of Eris Chric is stated:

"Set theory perhaps is too important to be left just to mathematicians."

I know several papers on connections between set theory and physics, but I don't know if these connections are important in physics. So my question is:

Question. Are there any applications of set theory in physics, which are of interest to physicists, and have important applications in physics?

• Links between Physics and Set Theory: sciencedirect.com/science/article/pii/S0960077996000550 a rather exhaustive discussion (with over 100 references) has been given by Augenstein in this paper, perhaps you'll want to focus your question a bit on one particular aspect? – Carlo Beenakker Mar 27 '15 at 11:14
• the journal may be controversial, but it's the author that counts, isn't it? en.wikipedia.org/wiki/Bruno_Augenstein --- many of these issues have also been addressed here mathoverflow.net/questions/27428/… and here mathoverflow.net/questions/10334/… – Carlo Beenakker Mar 27 '15 at 14:22
• Asking mathematicians whether or not some mathematics is important to physicists is perhaps a little un-natural. Why not ask physicists if you want a useful answer? – Ryan Budney Mar 27 '15 at 17:03
• IMO, at least 99 times out of 100, when someone says mathematicians are doing set theory wrong, that person is interested in some other structure, but wants to use the word "set" for the other thing. – Hurkyl Mar 28 '15 at 4:27
• It looks like sooner or later, every mathematician tries to look for applications of its work "to the real world". It is like suddenly they realize that they have spent so much time and effort developing something that may be useless for practical purposes, and they need to convince themselves that what they did actually matters. Young mathematicians don't get that, but when you put so much of your time into something, you end up wanting to actually matter for something, its normal. However, who knows what kind of maths will be needed in physics in 100 years? – Bilateral Mar 28 '15 at 12:09

Roger Penrose, in The Road to Reality, section 16.7 (size of infinity in physics) writes:

"...It is perhaps remarkable, in view of the close relationship between mathematics and physics, that issues of such basic importance in mathematics as transfinite set theory and computability have as yet had a very limited impact on our description of the physical world. It is my own personal opinion that we shall find that computability issue will eventually be found to to have a deep relevance to future physical theory, but very little use of these ideas has so far been made in mathematical physics".

I personally think that concept of time my be described better using objects in set theory: today we use real numbers as mathematical model for describing time, but maybe (some time!) physicist use a more complicated order, like objects we deal with them in set theory and other branches of mathematical logic, as a more accurate model for time. But, of course, its only an imagination!...

Arguably. modern research in set theory has made little impact elsewhere in mathematics. Even Replacement is rarely relevant, let alone large cardinals. Perhaps the stuation will change, as Harvey Friedman has predicted.

Meanwhile, work of Kristeva purports to apply the Generalized Continuum Hypothesis to poetry.

(I am not in any way casting aspersions on set theory, either as a discipline unto itself, or as an applicable field of mathematics. Rather, it is the rest of mathematics that needs to catch up. For starters, how about some applications of Replacement? Borel Determinacy is wonderful, but I would know people who might quibble about whether that result has implications for "mainstream mathematics".)

• I hope it's clear from what I wrote that I am not in any way casting aspersions on set theory, either as a discipline unto itself, or as an applicable field of mathematics. Rather, it is the rest of mathematics that needs to catch up. For starters, how about some applications of Replacement? Borel Determinacy is wonderful, but I would know people who might quibble about whether that result has implications for "mainstream mathematics". – Adam Epstein Apr 1 '15 at 19:47
• Replacement has tons of applications! From "having enough this-and-that" to the Reflection Principle which allows one to blindly work with proper classes in ZFC without resorting to Grothendieck universes and similar tools. The fact that replacement is invisible doesn't mean it's not used. This is a natural side effect that, while mathematicians conventionally work within the framework of set theory, most never work with bare axioms. In fact, that's the way set theory is intended to be used... – François G. Dorais Apr 3 '15 at 14:21
• @FrançoisG.Dorais Reflection Principles are unquestionably important, but still somehow metamathematical. Mostowski Collapse is tremendously useful in set theory, but irrelevant in settings where isomorphism invariance, rather than identity, is the issue. Regarding "enough injectives" in homological algebra, McLarty's recent work has eliminated transfinite recursion from Baer's argument. Do you have any examples of necessary use of Replacement in, say, algebraic topology/geometry/number theory, combinatorics, functional analysis, nonlinear PDE,...? – Adam Epstein Apr 3 '15 at 18:47
• By necessary I mean unprovable, or at least not currently known to be provable, in Zermelo set theory. – Adam Epstein Apr 3 '15 at 18:47
• In their book "Probability and Finance: It's Only a Game!", Shafer and Vovk use Borel determinacy to prove Kolmogorov's strong law of large numbers, certainly mainstream mathematics. – Michael Greinecker Aug 1 '16 at 23:09

Are there any applications of set theory in physics, which are of interest to physicists, and have important applications in physics?

I'm a physicist. Any answer to this question is going to depend completely on your definitions of "application" and "set theory." If you consider only some trivial corner of naive set theory, and count cases where its use is purely a matter of convenience or ease of notation, then certainly there are many applications. For example, we often solve quadratic equations in physics, and it's convenient to talk about the set of real solutions.

But every physics experiment that has ever been done was performed with finite physical and computational resources, which means that all of our experience of physics can be described within finitism. There are even rigorous arguments (Krauss 1999) that, given the cosmological facts we observe, any hypothetical future physical process will be able to harness only finite energy and finite computation. (This is nontrivial; before the discovery of dark energy, the opposite conclusion was reached by Dyson.)

Because of these physical limitations, it is not possible, even in principle, for us to measure an irrational number or to demonstrate affirmatively that spacetime has the structure of a manifold.

I only have access to the abstract of the Augenstein paper, but it seems to argue that "physical reality," specifically quantum mechanics, provides direct realizations of set-theoretical constructs, including "individual axioms" of ZFC. This sounds bogus to me on both physical and mathematical grounds.

Physically, Krauss's result tells us that we can never harness more than a finite amount of energy, and via the de Broglie relation, this puts a a cutoff on the wavelengths that we will be able to probe. The region within our cosmological event horizon will always be finite as well. The combination of these two cutoffs means that any quantum-mechanical experiment, even in principle, can be described by a finite-dimensional Hilbert space.

Mathematically, the vast majority of mathematics is carried out without any consideration of any underlying foundational issues, such as the use of ZFC as opposed to some other framework. The sphere within which physicists operate is even more restricted than that of normal mathematics.

We can also consider the role of computation, through which a physical machine (such as a computer, a brain, or a slide rule) can prove things about mathematics. I can use an analog computer to compute the square root of 2, e.g., by constructing two pendulums with lengths in a 2:1 ratio and measuring the ratios of their periods. If my analog computer says that the second decimal place of the decimal expansion of $\sqrt{2}$ is a 1, and your computation says that it's a 3, then my physics experiment has successfully demonstrated something to you about your mathematical theory. If the axioms of ZFC were to be directly realized in physical experiments, as Augenstein seems to propose, then I ought to be able to do the same kind of thing with ZFC. Does anyone really expect that a physicist will do an experiment that will prove the axiom of choice?

If you familiar with a bit of category theory and quantum mechanics then the following pretty paper written by Bob Coecke and Eric Paquette contain some interesting applications to quantum mechanics. For example, the inability to define a uniform copying operation in the category Set reflects the fact that we cannot copy unknown quantum states (the famous no cloning theorem in quantum mechanics).

There is an active field of research investigating foundational aspects in logic, computability and set theory and the connections to physics with applications to quantum information and computation, as well as in computational linguistics, computational logic and semantics.

• That doesn't have anything to do with (contemporary) set theory. Also, the category Set does have uniform copying and therefore allows for cloning, as discussed in Section 4.2 of the paper that you're referring to. – Tobias Fritz May 25 '16 at 19:57

I don't know how important these papers are considered to be, but they are by a physicist and published in a (mathematical) physics journal:

http://scitation.aip.org/content/aip/journal/jmp/17/5/10.1063/1.522953

http://scitation.aip.org/content/aip/journal/jmp/17/5/10.1063/1.522954

The idea of these papers is that a sequence of quantum measurements (or a single measurement of a continuous observable) produces a "random real", for a suitably defined notion of "random", and this implies that certain models of set theory are not good for quantum mechanics.