By Cantor, we know that there are different cardinalities of infinite sets. $\mathbb{N}$ is countable, $\mathbb{Z}$ is countable and - rather surprisingly - $\mathbb{Q}$ is countable, too. And even more surprisingly, $\mathbb{R}$ is uncountable.
I sometimes show Cantor's arguments in high school (where I teach) and they and their consequences are - indeed and expectedly - very surprising to learn about.
What are some other infinite sets which are surprisingly countable or uncountable?
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The set of natural numbers is countable as you have noted, also the set of all orderd pair's of numbers are countable (Intuitively this is because the size of this set is very close to the one of all rationals). Since a countable set A "generates" the countable set (A,A) (Thats not quite correct notation) we can use this argument to show that ((A,A),A)==(A,A,A) is alse countable, so any set of pairs or triples or millions of numbers is countable.
Cantor set, the set of points in $[0,1]$ whose decimal expansion has no prefixed number (for example, those nombers such that theres is no 5 in the decimal expansion).
All of them are, surprisingly (at least the first time one see them) uncountable. And the Cantor Diagonal Argument is key in proving this.