Comment by Viliam1234

I think the traditional word for the "super-natural number" is "nonstandard integer".

You correctly notice that in case of nonstandard integer, the recursive definition alone is ambiguous, because while the standard integers are connected to zero by a finite chain of successor operations, the nonstandard integers are only chained to each other in infinite chains unconnected to zero. So you could have multiple implementations of a recursive function, each of them giving the same value for the standard integers, but different values for the nonstandard ones.

But there is one extra constraint that I think you didn't take into account. Peano axioms contain the "axiom of induction", which... if you look at it from a certain perspective, says: "whatever (first-order statement) is true for standard integers, it must also be true for nonstandard integers". Well, it doesn't say that directly; it's more like "whatever is true/false for some integers, there must be a smallest integer for which it is true/false".

This further constrains the possibilities for the "+" operation. If you can e.g. prove for the standard integers that "a+b = b+a", then according to this axiom, the same must also be true for nonstandard integers. So if "nonstandard + standard" is defined unambiguously, then so is also "standard + nonstandard".

But this still leaves some space for ambiguity in defining "nonstandard + nonstandard".