if arithmetic is not axiomatizable, why are the Peano Axioms called so?

In mathematical logic it is proven that the theory of natural numbers is not axiomatizable nor enumerable, both in first order and second order logic.

Where axiomatizable means:

A theory (e.g. the theory of a structure) is axiomatizable if there is a decidable set of formulas whose set of consequences is equal to that theory

Why do we then still speak of the peano axioms?

Edit: it seems to me that there are three purposes of axioms:

1. use it to derive the theory of a particular structure such as the natural numbers.
2. distinguish a particular structure from all structures that are not isomorphic to it.
3. use it to distinguish certain categories of structures from others (e.g. structures that are groups from structures that are not groups).

Is that correct?

ps. see this post: Purpose of the Peano Axioms

So then perhaps the purpose of the peano axioms is nr. 2, instead of nr. 1.

• What do you mean not axiomatizable? – Tobias Kildetoft Sep 16 '16 at 8:46
• Because with Peano's axioms we can prove all "interesting" arithmetical facts: before G's Incompl Th, no unprovable arithmetical sentence was knonw, and in any case, the G's sentence is not "arithmetically interesting" at all... – Mauro ALLEGRANZA Sep 16 '16 at 8:53
• So it could be that some day, we find some arithmetical interesting fact that cannot be proven from PA? then we'd have to add a new axiom. By the way, what do you mean with "the G's sentence is not "arithmetically interesting" at all" – user56834 Sep 16 '16 at 8:56
• To your edit : Yes; Peano's original formulation was in what todays we call "second order" form. Thus, they are categorical. – Mauro ALLEGRANZA Sep 16 '16 at 10:00
• "the G's sentence is not "arithmetically interesting" at all" means that it is a formula of a unmanegeable lenght that is quite impossible to "produce" in practice and that - outside its "metamathematical reading" due to arithmetization of syntax - does not "means" nothing significant about natural numbers. – Mauro ALLEGRANZA Sep 16 '16 at 10:01