> Assuming The Natural Numbers are consistent then all true statements are provable in some axiom system. Just take the collection of all true statements as the axiom system. Now every true statement is trivially provable.
That axiom system wouldn't be particularly useful to humans though. When we talk about sets of axioms, we almost always talk about finite sets of axioms. This is what makes them useful to us, allows us to use them for describing things.
But you do have the right intuition here. The next step is using the compactness theorem[1].
It doesn’t have to be a finite set of axioms just recursively enumerable. The first order Peano axioms are not finite in number. One of them is an axiom schema. (I believe.)
> Assuming The Natural Numbers are consistent then all true statements are provable in some axiom system. Just take the collection of all true statements as the axiom system. Now every true statement is trivially provable.
That axiom system wouldn't be particularly useful to humans though. When we talk about sets of axioms, we almost always talk about finite sets of axioms. This is what makes them useful to us, allows us to use them for describing things.
But you do have the right intuition here. The next step is using the compactness theorem[1].
[1]: https://en.wikipedia.org/wiki/Compactness_theorem