Gödel's First Incompleteness Theorem states that if we have a recursive and consistent set of axioms $A$ in $\mathcal{L}_{\text{NT}}$, then there is a true first order statement about natural numbers $\sigma$ such that $A \not \vdash \sigma$.
Thus, the set of provable statements about the natural numbers is a proper subset of the set of true statements about the natural numbers. I was wondering what can be said about the set of provable statements with regards to its size in comparison with the set of true statements. Clearly they are both infinite sets, but, in terms of measure theory, topology, or density, is there something that can be said?