The uniqueness of the integers

It’s actually not too difficult to see that the integers are the only ordered integral domain with unit whose non-negative elements are well-ordered. So let’s go ahead and do it.

In fact, let’s try to build from the ground up. We can start with the additive identity and the unit . Since we’ve got an ordered ring we have to have , otherwise the multiplication can’t preserve the order.

Now we can also tell that is the smallest element larger than . Let’s say there were some element in between: . Then , and , and so on. The collection of all powers of has no lowest element, so the positive elements can’t be well-ordered in this case.

We can add up an arbitrary number of copies of to get , and we know there’s nothing between and , or else there would have to be something between and . We also get all the negative numbers since we have to have them. Multiplication also comes for free since it has to be defined by the distributive property, and every element around is the sum of a bunch of copies of .

Finally, the fact that we’re looking for an integral domain means we can’t introduce any relations saying two different elements like these are really the same in our ring without making a zero-divisor or collapsing the whole structure. I’ll let you play with that one.