The proof that I have in mind is as follows -
$\text{Gal }(\overline{\mathbb Q}/\mathbb Q)$ is a proper uncountable subgroup of the group of permutations on countably many symbols, hence the latter is uncountable. .
But it needs a lot of jargon from topology and algebra. Is there a neat proof ...
It is a nice question and the solution based on Riemann rearrangement theorem is very nice. (I certainly would not get an idea to use something like that.)
However, I was a bit surprised to see the question on MathOverflow. I would consider it a more-or-less standard freshmen exercise, at least the countable case.
There are also a few copies of this question on math.SE:
The proof that I have in mind is as follows -
$\text{Gal }(\overline{\mathbb Q}/\mathbb Q)$ is a proper uncountable subgroup of the group of permutations on countably many symbols, hence the latter is uncountable. .
But it needs a lot of jargon from topology and algebra. Is there a neat proof ...
