What I am wondering is if mathematicians know whether (assuming consistency) the natural numbers are a definite object, without ambiguity. This seems intuitively obvious, but I don’t know if its been shown. A formal question might be if two consistent axiomatic systems, each including all the axioms of the natural numbers and then some, can’t contradict each other in statements about the natural numbers in the language of the natural numbers (not referring to objects guarunteed by the other axioms of the systems).
The overwhelming consensus is that the true natural numbers is is a definite thing which objective properties — not because this can be proved from anything more basic, but because mathematicians intuitively believe they have some kind of objective Platonic existence.
It is known, however, that not all properties of the Platonic naturals can be captured completely by any reasonable axiomatic system. Given any axiomatic system $T$ that proves no falsehoods about the integers, Gödel’s incompleteness theorem shows how to construct an arithmetic sentence $\phi$ such that both $T\cup\{\phi\}$ and $T\cup\{\neg\phi\}$ are consistent theories.
A formal question might be if two consistent axiomatic systems, each including all the axioms of the natural numbers and then some, can’t contradict each other in statements about the natural numbers in the language of the natural numbers (not referring to objects guarunteed by the other axioms of the systems).
If “all the axioms of the natural numbers” just means Peano Arithmetic, then it is completely possible to have two axiom systems that are each consistent, each contain all of Peano arithmetic, and are inconsistent with each other.
This is because there are many statements that are not provable nor disprovable in Peano arithmetic, even though the statements are in the language of Peano arithmetic. Therefore, if $A$ is such a statement, “Peano arithmetic plus $A$” and “Peano arithmetic plus the negation of $A$” are an example of the phenomenon I just mentioned.
One reason that many mathematicians think that the natural numbers are well defined (even if the powerset of the natural numbers is not) is that the natural numbers are all about finiteness, which we may think we have a good sense about. There are really three phenomena that are interrelated:
The natural numbers
The set of all finite strings on the alphabet $\{0,1\}$
The set of all formal proofs from an effective set of axioms
Any of these concepts can be interpreted, in a certain sense, in any of the others. For example, if we know what a finite proof is, we can interpret a natural number as the length of a proof. If we know what a natural number is, we can define proofs in terms of their Goedel numbers.
So, if the natural numbers are somehow vague, the concept of a finite sequence and the concept of a formal proof must be equally vague. Most mathematicians think we have some sort of perception of these objects, although a minority think the concepts are vague.
One of the most interesting recent developments in this respect is the potential in multiverse set theory (as developed by Joel David Hamkins and coworkers) there is no “standard” model of the natural numbers, and that every model of set theory is not well founded relative to another model. I do not think anyone has yet written a philosophical argument about the consequences of that for foundations of mathematics.
HINT: Given two Peano systems $(N,0,S)$ and $(N’,0′,S’)$, prove there exists a bijection $f:N\to N’$ such that $f(0)=0’$ and $f(S(x))=S'(f(x))$.
This answer is perhaps related to that of Carl Mummert. There is an informal but very inspiring article by Joel David Hamkins, about his caveats or doubts about if we are correct in thinking that we have an absolute concept of the finite. if Hamkins have doubts about it, I am pretty sure they should be taken seriously. All other number theorists that believe that the standard model of Peano arithmetic defines THE natural numbers beyond any doubts, should not be taken too seriously.
You can read his inspiring thoughts at this link:
http://jdh.hamkins.org/question-for-the-math-oracle/
There is very little if any ambiguity in this case. For all practical purposes (putting aside Gödel), Peano’s axioms (2nd order) can be used as The Definition of the set of natural numbers:
$0\in N$
$S: N\to N$
$\forall x,y\in N:[S(x)=S(y)\implies x=y]$
$\forall x \in N: S(x)\ne 0$
$\forall P\subset N: [0\in P \land \forall x\in P: S(x)\in P \implies \forall x\in N: x\in P] $
It can be formally shown that if Peano’s axioms hold on set $N$ with successor function $S$ and first element $0$ (as above), then the system $(N,S,0)$ is unique to within an isomorphism, i.e. all such systems are identical except for the names used. See: http://dcproof.com/EquivalentPeanoSystemsB.htm
EDIT: Here is a nice visualization based on the falling dominoes paradigm of the kind of “junk terms” (e.g. the disconnected side-loop shown) that is excluded by the induction axiom (5).