Mathematicians define real numbers in an abstract way - as an 'ordered field' with 'the least upper bound property'. In physics, we use real numbers to represent distances. For us to be able to do this, we need to establish the following:
With every distance a real number can be associated.
With every real number, a distance can be associated.
In other words, a one to one correspondence. Is there any proof of this?
No. There is no proof of this.
What we do know is that the models that result from assuming it allow us to correctly predict the outcome of all experiments to date. Therefore, it is accepted for the same reason that most assumptions in physics are accepted: it works.
Your question seems to be: the reals as defined by mathematicians are characterized by a highly asbtract algebraic structure, so why should it have anything to do with distances?
Well the point of Physics is exactly that: we use mathematical structures to construct models of things that can be measured.
The field of real numbers is a mathematical structure. Distance, on the other hand, is something we can measure. In fact, it is in a sense defined by the way it is measured.
In the end your question is: why the field of reals as a mathematical structure is a good model for distances?
Now one possible answer is the following. How do we measure distances? Well, we pick something to call our fundamental unit of distance, then we take two locations $A$ and $B$, and we ask "how many copies of that object fit between $A$ and $B$?"
To be concrete, imagine you take a wooden stick as this fundamental object. Call it $\mathcal{W}$. It is clear we need $\mathcal{W},2\mathcal{W},\dots, n\mathcal{W}$ for any natural number $n\in \mathbb{N}$ to be able to measure possible distances. The counting property of the naturals make then a good model.
But it is not enough. In fact, take $A$ and $B$ to be $\mathcal{W}$ apart. There are many locations in between. What are their distances to $A$? We clearly need to be able to talk also about $\frac{n}{m}\mathcal{W}$. So fractions are also required to make a good model of distances. So we end up with $\mathbb{Q}$.
Now you might say: fine but why reals which now involve limits of Cauchy sequences of $\mathbb{Q}$?
The answer is: take 2 copies of your wooden stick. Place one in the horizontal and another in the vertical meeting at a location $O$, making up a $2$-dimensional cartesian system. Now, let $A$ be the location arrived at by walking one unit along each such direction. You end up in the point accross the diagonal of the square made by the wooden sticks.
Would it be reasonable the distance to this point $A$ be undefined? Well, in the $\mathbb{Q}$ situation it is. But if you go to the reals, then it is not. It is $\sqrt{2}\mathcal{W}$ distance from $O$.
So in the end $\mathbb{Q}$ is a somewhat poor model for distances, whereas $\mathbb{R}$ is a good one. So we use $\mathbb{R}$ for distances because it is a good model and using it we get predictions that can be checked and agree with observations.
By the way, in the international system of units (SI) this "standard distance" is called a meter.
There is no such proof. In fact this issue corresponds to what is often termed the "Ruler Postulate" in the parlance of the School Mathematics Study Group of the 1960-1970 era, and can be traced to Hilbert's axiomatization of Euclidean geometry, or that of Birkhoff.
It is necessary to postulate that the points on the geometrical (physical) line are in correspondence with the real number continuum as constructed mathematically. The ruler postulate is independent of other postulates. There are models where other Hilbert axioms hold true, but where the Ruler Postulate does not.
Physics makes models of reality. There is always some inaccuracy or approximation involved. We do not know a true, exact description of the universe.
There is more than one model. Classical physics works fine for most situations. When it doesn't, quantum mechanics or relativity may provide a more accurate answer. When they not are enough, there is qed or qcd.
The real numbers are used in all these models. We don't need proof here. We just define distances as real numbers or vectors.
The models do not perfectly represent reality or our ability to measure reality. E.G. Two real numbers can be arbitrarily close to each other or far apart. There are numbers so small or so big we cannot measure the corresponding distance. If you get below the Planck scale or above the size of the observable universe, we cannot even say if there are corresponding physically meaningful distances.
Other answers have touched on this, but I would like to make an explicit frame challenge.
For us to be able to do this, we need to establish the following: ... In other words, a one to one correspondence. Is there any proof of this?
Your question is using words that are not meaningful in this context. Proof is an abstract, mathematical thing that relates other abstract, mathematical things; it starts from a mathematical construction (a proposition), applies some mathematical transformations (proof steps, axioms, etc.), and arrives at a different mathematical construction (e.g., a truth value). Similarly, concepts like "one to one" are properties of mathematical objects like sets, functions, and groups.
So, asking if something abstract (like real numbers) can be proven to be "in one to one correspondence with" something physical (like distances) is a bit like asking if 1 "equals" an apple. It's not meaningful because they're different things. Math doesn't apply to the physical world. Which is of course why we have physics: to define some useful mathematical models of the real world that we can reason about and prove properties of.
So distance is defined in terms of real numbers, not because the universe is built of real numbers, but because operations on real numbers match with experimental evidence about how the universe behaves.
All that said, we've stuck with using real numbers to model distances because they seem to model reality "well enough" that we can continue to make useful predictions about it. And the fact that some branches of physics don't use real numbers in contexts where they don't make useful predictions (e.g., integers in particle physics and complex numbers in quantum mechanics) is in some sense the exception that proves the rule: if real numbers did not model distance closely enough, better predictions using other mathematical objects would motivate physicists to revise their models.
This is of course not a formal proof (because it can't be). We may discover things tomorrow for experiments that cause us to reevaluate our models. We may find that modeling distance with real numbers breaks down in some yet-undiscovered situations. But we can't know that until we find the experiment that shows it.
There are at least two senses of what we might mean here. Both of these have separate and distinct standards of proof, and for both, it is impossible to furnish such a proof. In both of them, the question comes down to, fundamentally, "what is the microscopic structure of the points on a line?" That is, "how can we describe all the points thereupon individually and their relationships to each other in an explicit, concrete manner based on indisputable constructs?" There are, in fact, mathematically, many, many different ways you can array such points and many different levels of detail at which you can talk of such arrangements which, makes this question all the more pertinent and definitely far from as obvious as you might at first think having been conditioned by teaching, drilling, and reprimand/"takemywordism".
We cannot provide a proof for either one. In the first case, different people may have different intuitions and, moreover, from a purely formally mathematical view, this thesis is like others such as the "Church-Turing thesis" regarding computation: it's basically an assertion of a standard position on the meaning of a term - here "distance" or "microstructure of the line", there, "computation" - that is inherently subject to contestability. Meaning is not something that can be proven/refuted: as the age-old question of prescriptivism in language illustrates.
For the second one, we can, at best, disprove this by showing that there existed, say, an absolute minimum distance scale. (The Planck length, by the way, is merely a proposal for such a scale - it is not by any means proven that this is and, in fact, there may be some evidence against it being so.) We cannot actually prove that the structure of physical space really is that of the real number line, or anything else that involves questions of what it looks like with infinite resolution, because all our empirical measurements can only ever have finite resolution if by nothing other than the simple fact that we cannot actually store the infinite amount of information that an infinitely precise measurement would represent. This means that there is no empirical way to distinguish physical space from being isomorphic to $\mathbb{R}$, or to a pixelated space with grain size $10^{-\mbox{Graham's number}}\ \mbox{US survey feet}$. Or, from the hyperreal and surreal numbers, or from more restricted but still rather complex systems like the computable numbers, which posit differences in detail at inaccessible infinitely fine scales. All we can surmise is that the space is at least as detailed and fine as the best measurements we have made so far.
In the end, $\mathbb{R}$ is a scientific model, just like all other parts of our physical theories are models, and that is used to represent physical space in all theories that are actually empirically validated. At no point should a model be assumed to be reality, but rather is a story we tell about and language we use to talk about, reality, and its empirical validation that it is a way of talking about reality that is faithful thereto in that it won't lead us to believe things happen that don't or that don't happen that do, to the extent thereof. There are likely many, many other stories we could tell thereabout that are just as good but for which historical contingency has effectively blinded us to.
And the reason we use $\mathbb{R}$ for building models is chiefly one of convenience: $\mathbb{R}$ is very nice to work with mathematically. Conceptually, it enjoys a simple structure: effectively, it can be considered as the natural result of wanting an integrated, streamlined number system in which you can talk about measurements at arbitrary levels of precision, up to a "clean" infinite precision which is useful for theoretical purposes. (Indeed, it is not too hard at all to hop from that to a formal definition or at least axiomatization.) Moreover, as such it ends up being very clean when it comes to formulating things like calculus, an indispensible tool of modern practice and physical and scientific model-building.
None of the other proposed alternatives to it - and there are many - have, so far, shown themselves to be quite as nice for model-building. If reality isn't so nice, fine, but even if we could somehow prove that, $\mathbb{R}$ would still continue to be very useful in working with approximate models for situations where we can ignore its true structure - e.g. pretty much all vocational and technological applications today. Even today, it is used in cases where we do know that the underlying phenomena aren't really like $\mathbb{R}$, e.g. many population growth models describe population as a real number, so one can avail oneself of tools like calculus in building them, even though of course we know that real populations of real organisms can only ever be whole numbers. $\mathbb{R}$ is literally that damn good.
After an exposition on how Descartes was the first modern mathematician to assert a one-to-one correspondence between numbers and line segments (noting that the Greeks might have had this idea too, but rejected it once incommensurables like the diagonal of a square were found), David Hestenes writes the following in the early pages of his book "New Foundations of Classical Mechanics":
A clear notion of "infinity" and with it a clear notion of the "continuum of real numbers" was not achieved until the latter part of the nineteenth century, when the number system was "arithmeticized" by Weierstrauss, Cantor, and Dedekind. "Arithmeticize" means define the real numbers in terms of the natural numbers and their arithmetic, without appeal to any geometric intuition of "the continuum". Some say that this development separated the notion of number from geometry. Rather the opposite is true. It consummated the union of number and geometry by establishing at last that the real numbers can be put into one to one correspondence with the points on a geometrical line. The arithmetical definition of the "real numbers" gave a precise symbolic expression to the intuitive notion of a continuous line (Figure 2.4).
That's on page 11 and the figure on page 10, which contains an elaboration of the mathematical logic he's alluding to. I think you can find it on Google Books. There are other interesting details in the section regarding Newton's thoughts on the relation between geometry, mechanics, and measurement (he says geometry is a subset of mechanics, and geometrical figures are what allow for measurement of the physical world), Euclid's thoughts on the relation between number and magnitude, and Descartes's role in developing and popularizing what you asked about, the one-to-one correspondence between the real numbers and line segments. I believe Descartes introduced this idea without formal justification and Hestenes notes that it was accepted without much hubbub by his contemporaries, likely because the notion of number had undergone a change in the preceding millennia that made the equivalence plausible.
What you're asking of course relates to what axioms we choose in making sense of the world and discovering what's true. You can't reason from nowhere. One could make a case similar to Newton's, that the notion of number is suggested to us by observing a world of differentiated objects, and the notion of magnitude or length is suggested by the geometrical form taken by the world and its objects, such that more of a justification is required in treating these objects as abstract and meaningful outside of a physical context!
So first off, what is a measurement? Well, using the example of length, I grab my measuring stick and lay it out until I reach the point I'm measuring to. Of course, it probably won't land on the point exactly. So I grab a smaller measuring stick and repeat with the remaining distance. That won't land exactly either, so I repeat the process until I decide it's good enough. I then stop there and call it a measurement.
Still, though, just because I decided it was good enough doesn't mean that's final. Maybe I left my glasses at home that day and couldn't judge it well, or maybe someone else comes along with a microscope and can see that it isn't, or maybe someone invents an even smaller measuring stick to use. With each improvement to our measuring device, the measurements should get closer and closer to each other (they form a Cauchy sequence). We would definitely like to think that any such sequence of measurements converges to the actual length of whatever we're measuring--that is, that all Cauchy sequences of lengths converge. This means that whatever kind of mathematical object lengths are, they form a complete metric space.
OK, but there are lots of complete metric spaces in mathematics. What else would we like to do with lengths? Well, let's take a closer look at that measurement procedure. It was pretty clear we need to be able to add lengths together when laying out our measuring sticks, and I don't think it's too controversial to suggest we should also be able to subtract them. This means lengths form a group. Further, we need to be able to talk about some lengths being longer than others in a way that sensibly interacts with adding them together. This means lengths are an ordered group. Lastly, it seems clear that we should only have to lay out our measuring stick a finite number of times for any sensible length. So lengths must also be an Archimedean ordered group.
While there are many mathematical structures that have some of these properties, there are only three Archimedean ordered groups that are also complete metric spaces: $\{0\}$, $\mathbb Z$, and $\mathbb R$.
So that's about the best we can do. It's hard to experimentally prove that $\mathbb R$ really describes the universe instead of being a very fine $\mathbb Z$. But we can at least be pretty sure that if measurements are meaningful, it's one of the two. And since $r\mathbb Z \subseteq \mathbb R$ for every nonzero real $r$, even if the universe is $\mathbb Z$, we can still use $\mathbb R$ to describe it if we like.
I heard that distances below the planck-length are physically meaningless, which would mean there‘s a correspondence between length and natural numbers rather than length and real numbers, implying that space is discreet rather than continuous.
I will not get into a discussion of the axiom of choice and the generalized continuum hypothesis (and the related work of Godel and Cohen in this direction), I leave that for the other answers. I want to mention though, that Godel believed that the power of the continuum (and I think he meant physical continuum) was aleph 2, not aleph 1. Since he also worked in physics (GR), not just in mathematical logic, I wonder why he believed that. The Lagrangian and Hamiltonian formalism (and the min action principle) are fundamental in physics. Every theory in physics (from QFT to GR) can be cast within this mathematical framework. If the power of the physical continuum is aleph k ( with k greater than 1), or lower than aleph 1 (in a universe where GCH is false), what would be the consequences in relation to the fundamental mathematical framework mentioned above? This question is closely connected to yours.
If a one-to-one correspondence between physical distances within a confined region of radius $r$ and the interval $[0,r]\subset\mathbb{R}$ were to exist in a physically meaningful way, then the region should contain an infinite amount of information. But the existence of an infinite amount of information in a region of finite volume and energy would violate the Bekenstein bound.
