How do we prove a set is uncountable? Are there some good examples?

Question

What is the general idea of constructing these proofs? Any links to good explanation are welcome. Thanks

Answer 1

There is no general way to show it but a simpler way to think about is to suppose it is true and show it leads to something impossible.

One trick used by Cantor ( to show that $\mathbb{R}$ is uncountable ) is, for example, to suppose that $\left[0,1\right]$ is countable. Yhen he founds a number that was not " counted ". You can see it here : https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument

Answer 2

The two most common strategies for proving a set $A$ is uncountable:

1. Assume we have a bijection $f$ between $N$ and $A$ and construct an element from $A$ that is not in the image of $f$. (Like in Cantor's diagonal argument) But there's no general way of doing that.
2. Construct a bijection $g$ between $A$ and some set that is known to be uncountable. (Like using $\tan$ to show that $[0,1]$ has the same cardinality as $\mathbb R$)