The natural numbers and their arithmetic properties are necessary for the study of any system in which objects may be repeatedly combined (or, more generally, mapped) in any way. Since the beginning of the 20th century, it has been customary to posit axioms which define the set of natural numbers. Here we will comment on some common approaches to this axiomatization.
First we must deal with an annoying but inescapable notational inconsistency. Is the smallest natural number 0, or is it 1 ? Fortunately, the texts for MATH 215, 231, and 303 all agree that the historical choice, N = {0,1,2,3,...} is the current choice. Another advantage of this choice is that the phrase "non-negative integer" is not needed, and the smoother "positive integers" then denotes the set {1,2,3,...}.
In our textbook "A First Course in Abstract Algebra", by Rotman, there is no intent to present a set of axioms for the natural numbers. All proofs are based on an assumption called the "Least Integer Axiom": "There is a smallest integer in every nonempty collection of positive integers". It is taken for granted that the underlying set {1,2,3,...} and the order relation are familiar. This is a perfectly reasonable "quick and dirty" method to get to the mechanics of inductive and number-theoretic proofs as quickly as possible. Given this as an axiom, it is very simple to prove the same property for N, explicitly: "There is a smallest integer in every nonempty collection of natural numbers." If this last statement is taken as an axiom, the first version is even easier to prove.
A true axiom system for the Natural Numbers would list a finite set of features of a set called N which guarantee that all 'known' theorems about the natural numbers may be proven about N given these axioms. As a philosophical aside, notice that, even with this attempted 'pure' approach, there is a correctness question: why should the theorems thought to be characteristic of the natural numbers by a particular consensus of mathematicians have absolute status? The answer is that no such status is really claimed.
Thus, in the period of the 1890's there was much discussion, if not debate, about the acceptability of a system proposed by Giuseppe Peano to define the natural numbers. Peano's Axioms were widely approved throughout the 20th century. Here's how they go:
There exists a set N and an injective mapping s from N to itself such that
It's not hard to guess that the mapping s is called the successor map and that, essentially, s(x)=x+1. Not too surprising, then, that the third axiom is called the axiom of induction!
Library Exercise: How do you define m+n, mn using s? How about "m < n"?
Of course, we need to be able to prove the Least Integer Axiom from Peano's if they are to be acceptable. This essentially comes down to assuming that the Principle of Induction is true and proving that every non-empty set of natural numbers has a least element. Here's a big hint in the form of a proof outline:
Argue by contradiction: assume we have found a non-empty set U of natural numbers which has no smallest element. Now prove by induction on n the following statement: For all positive integers n, if x is in U, then x is greater than or equal to n. Once that inductive proof is done, use the fact that U is assumed to be non-empty, so it must contain some number m. Now take n=m+1 and get an absurd inequality. So we couldn't have found such a U in the first place, and the proof by contradiction is complete.