I've been reading on set theory and I've found some interesting things.
First, the set of all natural numbers is countably infinite in cardinality. This infinity is denoted by $\aleph_0$. All good.
You can show a one to one correspondence between these natural numbers and integers in general, thus there are as many naturals as integers.
How is that possible?! If you can create a set similar to $\mathbb{N}$ from the negative values of $\mathbb{Z}$ then there is no way they can have the same cardinality. Unless of course, the notion of infinity in $\aleph_0$ dilutes this fact. Please elaborate.
Being infinite is equivalent, for a given set, to being in bijection with a proper subset. Thus the notion of infinity does dilute our intuition.
Cantor himself said "I see it, but I can't believe it", in a letter to Dedekind, referring to the existence of a bijection between $[0,1]$ and $[0,1]^2$.
For perhaps the simplest example of this phenomenon, consider the shift map, $x\to x+1$, which is a bijection of $\Bbb Z_{\ge0}$ with $\Bbb N$.