I skimmed that chapter before I posted my question. All I could find was information about what kind of coding is possible for arithmetizing syntax in $I\Delta_0$, but no standard theorems about combinatorics or number theory. Did I miss something?

As far as I understand, its original usage was for proving theorems about it, not for proving theorems within it. I'm thinking of the first-order theory whose axioms are those of PA without induction with the axiom that every number is either 0 or the successor of some number.

Thanks, I've removed the CH example in that case.

I didn't know that CH is a finite-order arithmetic sentence. Where on the arithmetic hierarchy does it lie?