I suspect most people trolling this subforum have Enderton's logic book.
Anyway. You're supposed to show that everything you can say about natural numbers with 0, successor, less-than, addition, multiplication, and exponentiation, has a proof from the axioms you're given. You need to do this inductively. Start with the two predicates = (from logic) and < and prove that for x and y of the form ,when x = y, this follows from the axioms, and when x y, this also follows from the axioms; then do the same for <.
To prove it works for anything with +, show it works for x + 0 = x (and fails for the rejection), then apply induction to 0. There's really a lot to do, though, so I don't want to go through every single step. The basic idea is you're providing a framework where (4+2) * 5 + 2 = 2^5 is proven true by the axioms (which it is), by changing (4+2) to 6, 6 * 5 to 30, 30 + 2 to 32, 2^5 to 32, and evaluates 32=32 as true...and also evaluates all false predicate resolutions as false. So you just need to translate the operations into terms that only have S and 0 in some effective fashion.