A simple example of an uncountable set that is not $\mathbb{R}$ 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 $\mathbb{R}$. 

One common proof technique to show that a set is uncountable is Cantor's diagonal argument. You can apply this to the example T. Bongers gave with infinite sequences of 0's and 1's.
Also, the powerset of any set is of a larger cardinality than the set. There is a good argument on Wikipedia.
