assumption of well-ordering principle

Question

An abstract algebra book says "the basic assumption here is 'well-ordering principle' : any nonempty set of nonnegative integers has a smallest member." and then using it, the book proves mathematical induction. Can we assume well-ordering principle because it is a theorem?

Answer 1

Yes, the well-ordering property of the integers follows from the construction of $\Bbb N$ (or $\omega$) in standard set theory (minimal inductive set, e.g.) It's a theorem of ZF.