What is a countable set?
In Wikipedia we read this definition:
In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers.
Now, what is the cardinality of $\mathbb{N}$? $\mathbb{N}$ contains an infinity of numbers so its cardinality should be $\infty$, isn't it? So a countable set could contain an infinity of elements? In this case, in which way is it countable?
Best Answer
If two sets are "of the same cardinality", that means that their elements can be paired off one-by-one against each other. As soon as we start doing this with a few different sets, we see that not all infinite sets are "of the same cardinality" in this sense. For example, $\Bbb R$ and $\Bbb N$ can NOT be paired off one-by-one against each other. In other words, there are different sizes of infinity.
The word "countable" just means that a set is EITHER finite, OR is of the smallest type of infinity (like $\Bbb N$). It's "uncountable" if it's a larger infinity - that is, it's too big to be paired off one-by-one against $\Bbb N$.