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This may seem like a naive/stupid philosophical question so I am prepared to get destroyed here but what is the real world relevance of the uncountability of the real number system? I understand that real analysis and as a result all sorts of applied mathematics rely on this property of the real line (and this may be the answer to my question), but can anything in the physical universe actually be uncountable? Or is this a concept that humans needed to create in order to rigorously develop the mathematics needed to measure/study things quantitatively in the real world? Thanks.

cushingtriad
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  • There might be some application, but I don't see why the lack of real world relevance make the uncountability of $\mathbb R$ an "artificial construct" – Rushabh Mehta Nov 10 '21 at 15:38
  • Point taken - artificial is the wrong word – cushingtriad Nov 10 '21 at 15:42
  • In your rewritten question, I think it is best to think of it as "Or is this a concept that humans needed to create in order to rigorously develop the mathematics needed to measure/study things quantitatively in the real world?" – Rushabh Mehta Nov 10 '21 at 15:43
  • Can anything in the physical universe be countably infinite? Can anything in the physical universe exceed the finite number $10^{100}$? What is the real-world relevance of the infinitude of the positive integers, or the existence of $10^{100}$? Shouldn't you ask those questions first? – MJD Nov 10 '21 at 15:45
  • One thing to note is that choice is provably equiconsistent with the rest of ZF. Highly infinitary objects are hard to interpret, but reasoning with them won't lead to error. Think of them as abstractions that stand in for limiting behaviors of finitary objects. – user3716267 Nov 10 '21 at 15:47
  • @MJD Depends what you mean by "anything" - if you count possible arrangements as "things," then you can get pretty huge numbers via combinatorics without much trouble. – user3716267 Nov 10 '21 at 15:49
  • Just a random thought: uncountability of real numbers makes it possible to define a translation-invariant Lebesgue measure on $\mathbb R^n$, which is close to saying "we can measure lengths, areas, volumes etc.". With a countable set we would need to either accept that some numbers have nonzero "weight" by themselves, or give up with $\sigma$-additivity. (Probably the second, but then a lot of calculus and probability theory also breaks down.) –  Nov 10 '21 at 15:57
  • Apparently continuity of spacetime as a "geometric object" is a theory that is being tested for validity, in the sense that some alternatives have been proposed, and that some of them have been falsified experimentally. For instance, apparently we have proof of the fact that modelling spacetime as a grid of size the Planck scale doesn't work (plus bonus). I don't know much about those things, but it appears that good ol' $\Bbb R^n$ is still alive and kicking. –  Nov 10 '21 at 16:34
  • Thanks for that link Saucy I was thinking about this same question the other day… – cushingtriad Nov 10 '21 at 18:30
  • And thanks for all the other instructive answers/comments – cushingtriad Nov 10 '21 at 18:31

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I think you have answered your own question: the real numbers are

a concept that humans needed to create in order to rigorously develop the mathematics needed to measure/study things quantitatively in the real world.

Note that the rational numbers suffice for recording all measurements.

I would replace "study" by "model". The invented mathematics of the mathematical continuum - calculus and its descendants - turns out to be extraordinarily useful at predicting the (rational) results of experiments.

That success says nothing about what the real world "is" in a philosophical or mathematical sense. It might be a large discrete object. We might be living in some kind of simulation ...

Ethan Bolker
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