Applications of uncountability of the real numbers 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.
 A: 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. 
A: 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. 
A: 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.
