“In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are a set of axioms for the natural numbers.
The constant 0 is assumed to be a natural number:
The next four axioms describe the equality relation.
2. For every natural numberx,x=x. That is, equality isreflexive.
3. For all natural numbersxandy, ifx=y, theny=x. That is, equality issymmetric.
4. For all natural numbersx,yandz, ifx=yandy=z, thenx=z. That is, equality istransitive.
5. For allaandb, ifais a natural number anda=b, thenbis also a natural number. That is, the natural numbers areclosedunder equality.
The remaining axioms define the arithmetical properties of the natural numbers. The naturals are assumed to be closed under a single-valued “successor” function S.
6. For every natural numbern,S(n) is a natural number.
Peano’s original formulation of the axioms used 1 instead of 0 as the “first” natural number. This choice is arbitrary, as axiom 1 does not endow the constant 0 with any additional properties. However, because 0 is the additive identity in arithmetic, most modern formulations of the Peano axioms start from 0. Axioms 1 and 6 define aunary representation of the natural numbers: the number 1 can be defined as S(0), 2 as S(S(0)) (which is also S(1)), and, in general, any natural number n as Sn(0). The next two axioms define the properties of this representation.
7. For every natural numbern,S(n) = 0 is false. That is, there is no natural number whose successor is 0.
8. For all natural numbersmandn, ifS(m) =S(n), thenm=n. That is,Sis aninjection.
Axioms 1, 6, 7 and 8 imply that the set of natural numbers contains the distinct elements 0, S(0), S(S(0)), and furthermore that {0, S(0), S(S(0)), …} ⊆ N. This shows that the set of natural numbers is infinite. However, to show that N = {0, S(0), S(S(0)), …}, it must be shown that N ⊆ {0, S(0), S(S(0)), …}; i.e., it must be shown that every natural number is included in {0, S(0), S(S(0)), …}. To do this however requires an additional axiom, which is sometimes called the axiom of induction. This axiom provides a method for reasoning about the set of all natural numbers.
9. If K is a set such that:
for every natural number n, if n is in K, then S(n) is in K,
then K contains every natural number.
The induction axiom is sometimes stated in the following form:
9. If φ is a unary predicate such that:
for every natural number n, if φ(n) is true, then φ(S(n)) is true,
then φ(n) is true for every natural number n.
In Peano’s original formulation, the induction axiom is a second-order axiom. It is now common to replace this second-order principle with a weaker first-order induction scheme. There are important differences between the second-order and first-order formulations, as discussed in the section Models below”
