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What applications are there of the theorem that the real numbers are uncountable?

I can list a few: A short proof that the irrational, non-algebraic and non-computable numbers exist. Any others?

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My interest in this question concerns an open problem: There is as of yet no proof that the real numbers are uncountable that doesn't use either Countable Choice or Excluded Middle. It's therefore interesting to see what this theorem is actually useful for.

wlad
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    Given any two points $(x_1,y_1),(x_2,y_2)$ in $\Bbb R^2$, you can find a path between them which does not cross any points $(p,q)$ where both $p,q$ are rational numbers. – JMoravitz Dec 10 '19 at 14:50
  • @JMoravitz what if $(x_1,y_1)$ is rational? – wlad Dec 10 '19 at 14:58
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    I'm not counting the endpoints of the path of course. Sketch of a proof... take the first point. Consider the set of lines going through that point. There are uncountably many of them as there are uncountably many choices for a slope. Of those, at most countably many will pass through a point $(p,q)$ with both $p$ and $q$ rational since $\Bbb Q^2$ is countable. Pick one that doesn't. Do the same for the other point and do so such that it is not parallel to the earlier line picked. These lines will then intersect. Traveling along the one line then along the other we have our path. – JMoravitz Dec 10 '19 at 15:02
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    Otherwise, you couldn't define Lebesgue measure, or the notion of Baire category, or many other notions that allow us to analyze sets of reals by discarding a "small" fragment and focusing on the rest, something which is quite common in analysis and topology. – Andrés E. Caicedo Dec 10 '19 at 16:12
  • @AndrésE.Caicedo This would make a good answer – wlad Dec 10 '19 at 16:14

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It gives a short argument for the fact that the vector space dimension of $\Bbb R$ over $\Bbb Q$ is infinite: $$ \dim_{\Bbb Q}(\Bbb R)=\infty. $$

Also, there is an application for computer science. Consider the set of functions that take an integer argument and return an integer result. This set is uncountable.
Since the set of computer programs is countable, there are uncomputable functions.

Dietrich Burde
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  • Regarding the second example, I'm specifically interested in the real numbers. Thanks – wlad Dec 10 '19 at 15:04
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In analysis and topology it is common to study sets of reals by classifying them according to some notion of "size". This is done when studying Lebesgue measure, or Baire category, or many other ("ideal-based") notions. Typically, in any of these contexts, we analyze sets of reals by discarding a "small" fragment and focusing on the rest. None of these notions would survive without the reals being uncountable. For instance, the countable union of singletons has measure 0, so any set of reals would have measure zero. This would make it impossible to develop the very useful integration theory of Lebesgue. Baire category is commonly used to establish existence results in settings where explicit construction of the desired objects is cumbersome or not feasible. Again, the countability of the reals would remove this approach from our toolbox.

Andrés E. Caicedo
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Not exactly an application, but we certainly don't want any mathematicians banging their heads against the wall...

It helped advance logic and formal methods. Once, almost a hundred and fifty years ago, Cantor demonstrated that the real numbers are uncountable, it begged the question

Find a set $S$ satisfying ${\displaystyle \aleph _{0}<|S|<2^{\aleph _{0}}}$.

Talk about something theoretically edgy!

And the ensuing lines of attack on this lead to us having to be accept (or reject) as an axiom the continuum_hypothesis.

CopyPasteIt
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