A *countably infinite* set is a set for which you can list the elements: $a_1,a_2,a_3,\ldots$.

For example, the set of all integers is countably infinite since I can list its elements as follows:$$0,1,-1,2,-2,3,-3,\ldots .$$So is the set of rational numbers, but this is more difficult to see. Let's start with the positive rationals. Can you see the pattern in this listing?$$\frac{1}{1},\frac{1}{2},\frac{2}{1},\frac{1}{3},\frac{2}{2},\frac{3}{1},\frac{1}{4},\frac{2}{3},\frac{3}{2},\frac{4}{1},\frac{1}{5},\frac{2}{4},\ldots .$$(Hint: Add the numerator and denominator to see a different pattern.)

This listing has lots of repeats, e.g. $\dfrac{1}{1}=\dfrac{2}{2}$ and $\dfrac{1}{2}=\dfrac{2}{4}$. That's ok since I can condense the listing by skipping over any repeats.$$\frac{1}{1},\frac{1}{2},\frac{2}{1},\frac{1}{3},\frac{3}{1},\frac{1}{4},\frac{2}{3},\frac{3}{2},\frac{4}{1},\frac{1}{5},\ldots .$$Let's write $q_n$ for the $n$-th element of this list. Then $0,q_1,-q_1,q_2,-q_2,q_3,-q_3,\ldots$ is a listing of all rational numbers.

A *countable set* is a set which is either finite or countably infinite; an *uncountable set* is a set which is not countable.

Thus, an uncountable set is an infinite set which has no listing of all of its elements (as in the definition of countably infinite set).

An example of an uncountable set is the set of all real numbers. To see this, you can use the *diagonal method*. Ask another question to see how this works.