I've seen it stated, for example here, that the integers are the unique commutative ordered ring with identity whose positive elements are well-ordered.
I understand why the integers are the smallest such ring, (and thus the initial object in the category of rings, for example), but why must they be the only one? In particular, why can't we have a commutative ring based on some larger well-ordered set? In particular, there are the "omnific integers" sitting within the surreal numbers, and presumably some hyperreal version of this as well. What I have in mind is some version of the natural operations on the class of ordinals, extended suitably (by the Grothendieck group construction perhaps, if the surreal number version doesn't work) to include additive inverses.
Why can't something like this work?
(Before you complain and say that surreals aren't a set, preventing them from being a ring, let's say that we define "the class of ordinals" by some suitable large cardinal or Grothendieck universe or something similar.)
EDIT As Wojowu pointed out in the comments, I'm probably over-thinking this: if $R$ is some commutative ordered ring whose positive elements are well-ordered, then there must be a map $f$ from $\mathbb{N}$ to $R$ by recursion (or initial-object stuff). Then, the set $X$ of all $x \in R$ which are greater than $f(n)$ for all $n \in \mathbb{N}$ is either empty (in which case it's not hard to verify that $\mathbb{Z} \simeq R$), or has a least element $x$. Then $x -1 < x$, so we have a contradiction.
Maybe there is a set vs. class subtlety here? If we take the actual class of all ordinals, then $X$ won't be a set, so we need well-ordering for sub-classes. What happens if we only assume well-ordering for subsets? Does this sort of ambiguity ever matter, or are the integers really that unique?
