## Peano Axioms

“In mathematical logic, the

Peano axioms, also known as theDedekind–Peano axiomsor thePeano postulates, are a set of axioms for the natural numbers.

The constant 0 is assumed to be a natural number:

- 0 is a natural number.
The next four axioms describe the equality relation.

2. For every natural number

x,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 number

n,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 asS(S(0)) (which is alsoS(1)), and, in general, any natural numbernasS^{n}(0). The next two axioms define the properties of this representation.7. For every natural number

n,S(n) = 0 is false. That is, there is no natural number whose successor is 0.8. For all natural numbers

mandn, 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 thatN= {0,S(0),S(S(0)), …}, it must be shown thatN⊆ {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 theaxiom of induction. This axiom provides a method for reasoning about the set of all natural numbers.9. If

Kis a set such that:

0is inK, and- for every natural number
n, ifnis inK, thenS(n) is inK,then

Kcontains every natural number.The induction axiom is sometimes stated in the following form:

9. If φ is a unary predicate such that:

- φ(0) is true, and
- for every natural number
n, if φ(n) is true, then φ(S(n)) is true,then φ(

n) is true for every natural numbern.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”

(Source: *Wikipedia*)