# Infinite Sets (real rational integers) [duplicate]

How can the real, integer, and rational number sets be infinite, yet, they aren't all the same size?

## marked as duplicate by user228113, Community♦Dec 16 '15 at 22:25

Given the definitions of what it means for two infinite sets to be of the same size, and of what it means for one set to be at least as large as another, $\Bbb N$ and $\Bbb Q$ are of the same size, because we can prove that there is a bijection between them.
However, Cantor's theorem shows that no 1-1 function $\Bbb N\to \Bbb R$ can exhaust all of $\Bbb R$ — it can't be onto $\Bbb R$, so $\Bbb N$ is strictly smaller than $\Bbb R$.