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 $\mathbb{R}$. 

The set of all partitions of $\Bbb N$. In fact, the set of all partitions of $\Bbb N$ into two parts is already enough.
