system
Given the following post, respond as if you're commenting on Reddit:
human
Am I correct in thinking zero is the middle of all the numbers?

Or are there any weird high-end math type things that keep this from being true?
gpt
Well, it depends entirely on what you mean by "in the middle of the numbers".

One possible definition of "in the middle" is this: *A number x is in the middle of all numbers if and only if there is a one-to-one pairing between the numbers smaller than x and the numbers larger than x*.

Before we go on, let's try our definition out and see if it makes sense. Let's consider just these five numbers: 1, 2, 3, 4, 5. Now, is 2 in the middle of these numbers? Let's see. There's only one number less than 2 (namely, 1), but there are three numbers greater than 2 (namely, 3, 4, and 5). There is no way to make a one-to-one pairing between the set {1} and the set {3, 4, 5}, so 2 is not in the middle of these numbers.

Okay, is 3 in the middle of these numbers? There are two numbers less than 3 (namely, 1 and 2), and there are two numbers greater than 3 (namely, 4 and 5). It is possible to make a one-to-one pairing between {1, 2} and {4, 5}: for example, we can pair 1 with 4, and 2 with 5. Therefore, 3 *is* in the middle of these numbers.

Now, it may seem like we've made things needlessly complicated. The point of mathematics, though, is to make things absolutely precise and exact. As a result, mathematics allows us to know things with absolute certainty.

Okay, so, if we use this definition, then is 0 in the middle of all numbers? Yes, it is. There are infinitely many numbers smaller than 0 (-1, -2, -3, ...) and infinitely many numbers larger than 0 (1, 2, 3, ...). We can pair -1 with 1, -2 with 2, -3 with 3, and so on, thus making a one-to-one pairing.

There's an interesting consequence of our definition, though: it states that *every* number is in the middle. Let's look at an example here.

Is 10 in the middle of all numbers? There are infinitely many numbers less than 10 (9, 8, 7, ...) and infinitely many numbers greater than 10 (11, 12, 13, ...). It's possible to make a one-to-one pairing between these two sets: pair 9 with 11, 8 with 12, 7 with 13, and so on. So yes, 10 is in the middle.

The important thing to realize is that it depends on how you define "in the middle". It would be possible to come up with a different definition of "in the middle" such that 0 is in the middle, and nothing else is in the middle; or yet another definition of "in the middle" such that nothing at all is in the middle.