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