# Countability in the Integers [closed]

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?

• I'm a bit confused by the question, but $\infty$ and $-\infty$ are not integers (nor do ordinal numbers come into play, really) - hence why Rudin would not consider them. – Milo Brandt Feb 17 at 5:03
• same cardinality means they can be put in one-to-one correspondence – J. W. Tanner Feb 17 at 5:04

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.

• -- thanks people – Lol Flo Feb 17 at 6:15
• I have just tried reading it again and I lost track, because there isn't infinity this time. If there are twice as many integers than positive integers, how is there a 1 : 1 correspondence when the terms seemingly run out? – Lol Flo Feb 17 at 7:25
• @LolFlo Where do they run out? The 1-1 correspondence in this answer shows that there aren’t twice as many. – amd Feb 17 at 7:33
• For example, how is counting possible after counting the positive numbers first? – Lol Flo Feb 17 at 7:36
• I'll rephrase - picture the negative integers have cardinality equal to the positive integers. The positive integers also have that cardinality. It doesn't add up for me! – Lol Flo Feb 17 at 9:18