[johnbender]

I will try to translate my understanding:

> This collection forms an axiom system for the Natural numbers and all true statements are provable in this system. Indeed, every true statement is an axiom.

You have defined the axiom system as the set of all true statements about natural numbers so of course all true statements are provable! But crucially ...

> The problem with doing this is that the collection of all true statements about the Natural numbers is not recursively enumerable.

We want an extensional definition of a set of axioms. One way to get that is to enumerate it! This is important because ...

> Recursively enumerable was the goal of Hilbert and others because they wanted to reduce mathematics to an algorithmic or mechanical process of verification.

I can sympathize with this goal being that I use a proof assistant quite frequently.

Thank you for taking the time to respond!

[yequalsx]

You're welcome. I think your understanding is correct.

I will add one more thing for completeness sake. There are statements that are true of the natural numbers, the natural numbers that you and I think of when see this term, that can't be proven in the first order theory. This means that there is a non-standard model in which there is a non-standard integer that is a counterexample to the statement. The second order axioms are categorical so there is a proof of such a statement using the second order axioms. One may not be able to find such a proof but there is one.