It's been 20+ years, and I don't remember the exact steps the course went thought. But basically we started with defining N using the Peano axioms (although I don't recall the name 'Peano' being mentioned) and proving some basic rules of addition and equality. Then we defined subtraction, multiplication and inverse elements, and constructed Z and Q. From there you get to algebraic groups, and from there you can hand wave a lot of details.
As I said it was literally one of the first math courses we did, so it wasn't super rigorous with all the technical and logical details.