According to Gödel's incompleteness theorem, there exists a sentence $G$ in the vocabulary of number theory ($N$) which is not provable from any (recursively enumerable) consistent set of axioms $T$, and yet which is true under the standard model ($SM$) of $N$. Also, this means that there is some other model $M$ of $T$ under which $G$ is false (for otherwise, if $G$ was true in all models of $T$, then $G$ would be provable from $T$, by the completeness theorem for FOL).
My question is, how, or in what fashion, do we specify or describe the standard model $SM$ of $N$? Some kind of a description/specification (in some metalanguage perhaps) seems necessary if we are to show that $G$ is true under $SM$, and also if we are to differentiate it from some other, non standard model $NSM$ of $N$. But once we have such a formalized specification $S$, isn't that itself a kind of an axiomatization of the standard model (which presumably can be expressed in first order logic)?
In other words, from S we would be able to prove any sentence $\phi$, as long as $\phi$ is valid under all models $M$ described by $S$ (i.e. $M\models S$). But since S is a formalized description/specification of the standard model of $N$, the models described by it have exactly the structure of what we normally mean by "numbers" and "arithmetic" and "number theory", and hence we would be able to prove any number theoretical sentence that was truly of interest to us.