Countability in the Integers

Question

In Rudin's principles of mathematical analysis in chapter 2, it says that the integers are countable as a set because of having the same cardinality as the positive integers. However it's just not making sense because, as the paper counts out the integers $0, 1, -1, 2, -2...$ I think of adding in $\infty$ and - $\infty$. In the positive integers to match that I add in $\infty$ plus an ordinal number higher than that one; that induces another step of adding another ordinal to the integers and the positive integers end up with 1 more cardinality. What is the mistake?

Answer 1

The set of integers is numerable because you can put it into a one to one correspondence. For instance you can

1. $0\to 0$
2. $1\to +1$
3. $2 \to-1$
4. $3 \to +2$
5. $4 \to -2$

and so on.