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? 

 A: 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!... 
A: 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".)
A: 
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?
A: 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.
A: 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.
