TITLE: A simple example of an uncountable set that is not $\mathbb{R}$
QUESTION [24 upvotes]: Let's suppose that I have only defined $\mathbb{N}$ and then I define the terms finite and infinite set, and also countable and uncountable set. 
I can think of some examples of finite, infinite and countable sets, but what about uncountable sets? I think the simplest example is $\mathbb{R}$. But as I don't have it defined yet then I cannot use it. 
What may be a good and simple example in this case?. By simple I mean easy to verify without knowing anything about ordinals, cardinals or $\mathbb{R}$.

REPLY [4 votes]: As everyone said, P(N), which is the set of all subsets of N is uncountable. I'm just gonna add the proof.
You can first use this theorem 3.3 and its proof in the link below:
http://www.math.psu.edu/wysocki/M403/Notes403_3.pdf
As there is not any surjective map between any set A and P(A) from the theorem above, then there is no surjective map between N and P(N).
And as "countable set" is defined as "If a set C is a countable set, then there exist a map between C and N which is 1-1 and onto (surjective)"; then P(N) is not countable, as there is no surjective map from N to P(N).
q.e.d.