in the familiar language of decimal representation) is the multiplicative right identity: To show that is also the multiplicative left identity requires the induction axiom due to the way multiplication is defined: is the left identity of 0: . If is the left identity of (that is ), then is also the left identity of : . Therefore, by the induction axiom is the multiplicative left identity of all natural numbers. Moreover, it can be shown that multiplication is commutative and distributes over addition: . Thus, is a commutative semiring. Inequalities The usual total order relation ≤ on natural numbers can be defined as follows, assuming 0 is a natural number: For all , if and only if there exists some such that . This relation is stable under addition and multiplication: for , if , then: a + c ≤ b + c, and a · c ≤ b · c. Thus, the structure is an ordered semiring; because there is no natural number between 0 and 1, it is a discrete ordered semiring. The axiom of induction is sometimes stated in the following form that uses a stronger hypothesis, making use of the order relation "≤": For any predicate φ, if φ(0) is true, and for every , if φ(k) is true for every such that , then φ(S(n)) is true, then for every , φ(n) is true. This form of the induction axiom, called strong induction, is a consequence of the standard formulation, but is often better suited for reasoning about the ≤ order. For example, to show that the naturals are well-ordered—every nonempty subset of N has a least element—one can reason as follows. Let a nonempty be given and assume X has no least element. Because 0 is the least element of N, it must be that . For any , suppose for every , . Then , for otherwise it would be the least element of X. Thus, by the strong induction principle, for every , . Thus, , which contradicts X being a nonempty subset of N. Thus X has a least element. Models A model of the Peano axioms is a triple , where N is a (necessarily infinite) set, and satisfies the axioms above. Dedekind proved in his 1888 book, The Nature and Meaning of Numbers (, i.e., “What are the numbers and what are they good for?”) that any two