infinity

What does it mean to count? When I ask you “How many apples are there” depending on however many apples you are counting, you’ll mentally provide some number. $1,2,3,4,\ldots$ etc. Counting is counting, you’ve done it since you were a child.

The Rationals and the Reals On the previous page we showed (kinda) how the rationals $\mathbb{Q}$ was a countable set the same size as the naturals $\mathbb{N}$. Right now we will take some time to how $\mathbb{Q}$ is quite intertwined with the set of real numbers $\mathbb{R}$.