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:
- use it to derive the theory of a particular structure such as the natural numbers.
- distinguish a particular structure from all structures that are not isomorphic to it.
- 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.