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What are some topics in real analysis that make use of infinite cardinalities larger than that of the real numbers themselves, preferably [edit: but not necessarily] topics that are widely applied in scientific applications?

Edit: While there may be not any actual infinities in science, the notion of infinity plays an essential part in real analysis which is indeed widely applicable in science, e.g. in the notion of a limit, the Fundamental Theorem of Calculus and the Intermediate Value Theorem . I was just wondering if higher orders of infinities are similarly required.

Dan Christensen
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    "widely applied in scientific applications" ... There are none. – William Sep 22 '14 at 04:27
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    The question in the title and the question in the body are different. – Asaf Karagila Sep 22 '14 at 04:28
  • If you assume there is a measurable cardinal (which is very large), then analytic and coanalytic subsets of the reals behave more nicely. – William Sep 22 '14 at 04:45
  • @William Not sure what you are getting at, but how widely applicable is this notion in science? – Dan Christensen Sep 22 '14 at 04:49
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    You are aware that science is finitary, right? Theoretical mathematics dealing with infinite objects don't have direct applications to science, and surely not "widely applicable". – Asaf Karagila Sep 22 '14 at 05:58
  • Relevant to the question in the body of the post: one, two, and three (see my answer on the third). Relevant to the question in the title this and the links there might be related, also that. – Asaf Karagila Sep 22 '14 at 06:33
  • @AsafKaragila Even though science may itself may be finitary, the infinite plays an essential part in topics in real analysis that are indeed widely applicable. I was hoping to find similar topics for which higher orders of infinity are necessary. – Dan Christensen Sep 22 '14 at 14:07
  • Dan, you can replace all those appearances of infinity by a sufficiently large finite number, and science would work just fine. The only reason we use infinity is that we don't want to limit what finite number we choose. But the notion of infinity from real analysis is not one of cardinality. So asking this question is both irrelevant to the infinity from real analysis, as well confusing since the title and the body talk about completely disjoint questions. – Asaf Karagila Sep 22 '14 at 14:11
  • @Asaf: You can replace all appearances of a complex number with a pair of real numbers and science (and math) will work fine too. that doesn't mean that doing so is a good idea and that there aren't applications of complex numbers. –  Sep 22 '14 at 14:21
  • @Hurkyl: Calling complex numbers by a different name won't change them. Taking "limits" over an essentially discrete set (where the differences are just incredibly small, say $10^{-1000000}$ between the points) which is also bounded (by a very large number) is an actual difference from talking about infinity. I have no intention of arguing over this point, though. – Asaf Karagila Sep 22 '14 at 14:24
  • Dan, the notion of infinity in calculus is not the same infinity as infinite cardinals or infinite ordinals. Your question about infinite cardinalities is misplaced. – Asaf Karagila Sep 22 '14 at 14:25
  • @AsafKaragila The Fundamental Theorem of Calculus and the Intermediate Value Theorem are applied everywhere in physics. I don't see how these could be got at in essence using a "sufficiently large finite number." – Dan Christensen Sep 22 '14 at 14:26
  • @Dan: And yet the notion of "quantum" is discretization of the space. – Asaf Karagila Sep 22 '14 at 14:27
  • @AsafKaragila There are still interesting applications of the infinite in Newtonian mechanics, which itself, is still widely applicable. – Dan Christensen Sep 22 '14 at 14:32
  • An particle on the other side of the universe is only a finite distance away from me, but if you calculate the gravitational pull it exerts on our planet you will find it infinitesimally small. Sufficient to be neglected. And lo how many things scientists discard for being so tiny. My point is that you can read that level of "tiny" at a finite stage, and so you don't have to use infinities. In either case, this infinity is NOT A CARDINAL NUMBER, and application of uncountable cardinals to real analysis are not "widely applicable in science". – Asaf Karagila Sep 22 '14 at 14:35
  • @AsafKaragila How do you mathematically describe path of particle falling to the ground the ground (assume constant $g$) without implicit use of IVT? – Dan Christensen Sep 22 '14 at 14:49
  • The same way you can describe the change of a price of a stock at the closing time each day over a year. Discretely, over a quantized unit of time. And if there are more than 25 frames per second, then my mind will not be able to tell the difference between that and a continuous movement. – Asaf Karagila Sep 22 '14 at 14:52
  • @AsafKaragila How would you obtain the result that the time $t$ to fall a distance $h$ is given by $\sqrt{2h/g}$ without implicit use of FTC or IVT? – Dan Christensen Sep 22 '14 at 15:05
  • @Asaf: To the best of my knowledge, quantum geometry is still purely theoretical -- and even then, actual quantum states would still be superpositions of various configurations with continuous coefficients. –  Sep 22 '14 at 15:18
  • @Hurkyl: Lots of things are theoretical. Video, however, is proof that discretization is sufficient. Dan, it's like you don't listen. So I'm gonna stop talking. Do note, again, and again, and again, that the infinity in real analysis is not a cardinal number. – Asaf Karagila Sep 22 '14 at 15:25
  • @Asaf: Discretization comes with side effects -- even video is clearly insufficient, as anyone who has tried to video-tape a television screen knows. –  Sep 22 '14 at 15:28
  • @Hurkyl: Yes, but that is because of the scanning effect of CRT screens. On LCD/LED based screens you see flickering. But even cameras can be fooled by fast enough objects, and faster cameras will be fooled by even faster objects. I didn't say that just 25 frames per second, loaded sequentially one at a time, is the correct way. But I pointed out that in our mind, 25 frames per second suffice. And in the "eyes" of cameras some larger number would suffice. So calculating everything within frames of a fraction of a second, or some other incredibly small frame (e.g. Planck time) should suffice. – Asaf Karagila Sep 22 '14 at 15:32
  • @Asaf: As I recall, even that's not fine enough: theories predict that we should see blurring of distant objects, and other effects like high energy photons outracing low-energy photons. But sure, you can suppose an even finer scale, but ultimately all you're doing is postulating a discrete approximation to the continuum and isn't really different at all from replacing complex numbers with pairs of real numbers. –  Sep 22 '14 at 15:41
  • @AsafKaragila Just to confirm your position: Is it correct to say that, in your opinion, infinite cardinals of any size play no role whatsoever in real analysis? – Dan Christensen Sep 22 '14 at 15:43
  • @Dan: I'm pretty sure he means that cardinality really has little or nothing to do with things like continua, or things like $+\infty$ that appear in limits that the OP alluded to. –  Sep 22 '14 at 15:44
  • @Hurkyl: Dan is the OP (unless you mean original post, not original poster). Dan: You could argue that some things in analysis might mention cardinalities. E.g., a monotone function can only have countably many discontinuities. But this is not the same as $\pm\infty$ which you refer to in the post and in all the examples you gave in the comments. – Asaf Karagila Sep 22 '14 at 16:26

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The requirement "widely applied in scientific applications" is probably too high of a bar to get anything of interest, but I can think of several examples in real analysis where cardinalities higher than $2^{{\aleph}_0} = c$ are applied.

The standard argument that there exists a Lebesgue measurable set that isn't a Borel set is an example: The Cantor middle thirds set has $2^c$ many subsets, all of which have Lebesgue measure zero, but there are only $c$ many Borel sets.

A similar argument shows there exist Lebesgue measurable functions that are not Borel measurable: There are $2^c$ many characteristic functions of subsets of the Cantor middle thirds set, but there are only $c$ many Borel measurable functions.

Miroslav Chlebík, in this 1991 Proc. AMS paper, showed there exists $2^c$ many symmetrically continuous functions from the reals to the reals, and thus there exist symmetrically continuous functions that are not Borel measurable. When Chlebík's paper appeared, it had been a long unsolved question whether there even exists a symmetrically continuous function that isn't a Baire one function. See also the math StackExchange question Does $\lim_{h\rightarrow 0}\ [f(x+h)-f(x-h)]=0$ imply that $f$ is continuous?.

Because the union of interior of the unit disk in ${\mathbb R}^2$ with any subset of its boundary is a convex set, there exist $2^c$ many convex sets in ${\mathbb R}^{2}.$ Since there are only $c$ many Borel subsets of ${\mathbb R}^{2},$ it follows that there exist convex sets in ${\mathbb R}^2$ that are not Borel. (Note that this is so not true in ${\mathbb R}.)$

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Dave L. Renfro
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  • Yes, those are nice applications of cardinals larger than $\frak c$ to real analysis. Their application to "science" remains to be seen. – Asaf Karagila Sep 23 '14 at 20:30
  • @Asaf Karagila: And thus the point of my first paragraph... Incidentally, I started writing this as a comment to Nick R's answer, but then decided it was getting too long (and might be of wider interest), so I made it an answer. – Dave L. Renfro Sep 23 '14 at 20:36
  • Oh yeah, I am aware of that. I was just explicitly stating this, since the thread seem to have been started by some misguided notion that "widely applicable in science" has a nontrivial intersection with "infinite" (in the sense of cardinality). – Asaf Karagila Sep 23 '14 at 20:40
  • @Asaf Karagila: I could have also said (and now I am saying) that I decided to answer the question that should have been asked. As for the nontrivial intersection, this also holds for most areas of real analysis I'm interested in. – Dave L. Renfro Sep 23 '14 at 20:47
  • @DaveL.Renfro Thanks Dave (+1). Is it fair to say that, in those topics of real analysis that are widely applied in science and engineering, discussions of orders of infinity are confined to sets being either countable or uncountable? That is my recollection of many years ago. – Dan Christensen Sep 26 '14 at 14:47
  • @Dan Christensen: I would agree, even to the point that such discussions rarely come up. Maybe separability of some of the better known function spaces would be an example, but when I think of science and engineering applications, I think of things like identities and properties of the gamma function and of the elliptic functions, and convergence issues (often with estimates included) for sequences of points and for sequences of functions, and other fairly concrete issues. – Dave L. Renfro Sep 26 '14 at 14:57
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My modest understanding of real analysis leads me to believe that infinite cardinals have no role to play here. Indeed, the thrust of the subject's development was to replace the actual infinities of its original formulation with the potential infinities we use today in expressing the notion of convergence and limits.

Regarding scientific applications, the finite nature of our science seems to imply that any role infinity may take can only be the result of our mathematical idealization and abstraction.

gamma
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  • Thanks, Nick. I was just wondering if any such "mathematical idealizations and abstractions" employed in science made use of any use of cardinalities greater than that of the real numbers themselves -- other than, of course, simply admitting the possibility of their existence, e.g. $2^{\mathbb{R}}$. – Dan Christensen Sep 22 '14 at 18:13