You need to notice that the standard decimal representation of the natural numbers is not a representation from inside the Peano model. The language only contains a constant symbol for $0$. Then one introduces $1=S(0)$ as shorthand notation not as a new symbol. Then comes $2=S(1)=s(S(0))$ again a...
it's just introducing shorthand notation for strings in the language. You could have used "banana" as shorthand for S(S(S(S(S(S(S(S(S(0)))))))))) if you want. But it makes more sense for us humans (who like numbers and use the decimal notation regularly) to call it something a bit more meaningful. We work with the model of PA from outside of it. We are not in the model so we are free to study it by means of things we have outside as well. It makes life easier.
@GitGud recursively: n+1 is shorthand notation for S(n), where 0 is shorthand for "0". It works since you believe that for the models of the naturals that you have in your head, recursion works.
@RustynYazdanpour In order to talk about the set $\{ 0, 1, 2, 3, 4, \dots \}$ we must previously have some meaning for $1, 2, 4, 5, 6, 7, 8, 9$, which I've done in my question. But then what meaning do you give to the symbol $10$? And to the symbol $100000000000\cdots 001$? I want to be able to answer this question to any sequence of the symbols $1, 2, 3, 4, 5, 6, 7, 8, 9$ and $0$.
@IttayWeiss So $9+1$ is $S(9)$, i got no problems with that because I know what $1$, $+$ and $9$ are. But I don't know know that $10$ is $S(9)$ or $9+1$.
9+1 is S(9), which some people call "ten", how you represent it typographically is an irrelevant question. Any typographical representation will be shorthand for a string in the language. These strings are defined recursively as above and do not require any representation of n.
How can you call it irrelevant if that's how you think about natural numbers? I don't agree that the way we typographically represent mathematical objects is irrelevant.
@IttayWeiss Deep results? I wouldn't know about that. But basic summation, multiplication and division rely on the typographical representation. I don't consider that irrelevant. And even if is, my question still makes sense, doesn't it? This discussion doesn't belong here anyway...
Discussion between Ittay Weiss and Gi…
Discussion between Ittay Weiss and Gi…