## Peano axioms |

In **Peano axioms**, also known as the **Dedekind–Peano axioms** or the **Peano postulates**, are

The need to formalize ^{[1]} In 1881, ^{[2]} In 1888, *The principles of arithmetic presented by a new method**Arithmetices principia, nova methodo exposita*).

The Peano axioms contain three types of statements. The first axiom asserts the existence of at least one member of the set of natural numbers. The next four are general statements about ^{[3]} The next three axioms are **Peano arithmetic** is obtained by explicitly adding the addition and multiplication operation symbols and replacing the

- formulation
- arithmetic
- first-order theory of arithmetic
- models
- nonstandard models
- consistency
- see also
- notes
- references
- further reading
- external links

When Peano formulated his axioms, the language of * Begriffsschrift* by

The Peano axioms define the arithmetical properties of * natural numbers*, usually represented as a

The first axiom states that the constant 0 is a natural number:

- 0 is a natural number.

The next four axioms describe the ^{[5]}

- For every natural number
*x*,*x*=*x*. That is, equality isreflexive . - For all natural numbers
*x*and*y*, if*x*=*y*, then*y*=*x*. That is, equality issymmetric . - For all natural numbers
*x*,*y*and*z*, if*x*=*y*and*y*=*z*, then*x*=*z*. That is, equality istransitive . - For all
*a*and*b*, if*b*is a natural number and*a*=*b*, then*a*is also a natural number. That is, the natural numbers areclosed under equality.

The remaining axioms define the arithmetical properties of the natural numbers. The naturals are assumed to be closed under a single-valued "*S*.

- For every natural number
*n*,*S*(*n*) is a natural number. - For all natural numbers
*m*and*n*,*m*=*n*if and only if*S*(*m*) =*S*(*n*). That is,*S*is aninjection . - For every natural number
*n*,*S*(*n*) = 0 is false. That is, there is no natural number whose successor is 0.

Peano's original formulation of the axioms used 1 instead of 0 as the "first" natural number.^{[6]} This choice is arbitrary, as axiom 1 does not endow the constant 0 with any additional properties. However, because 0 is the *S*(0), 2 as *S*(*S*(0)), etc. However, considering the notion of natural numbers as being defined by these axioms, axioms 1, 6, 7, 8 do not imply that the successor function generates all the natural numbers different from 0. Put differently, they do not guarantee that every natural number other than zero must succeed some other natural number.

The intuitive notion that each natural number can be obtained by applying *successor* sufficiently often to zero requires an additional axiom, which is sometimes called the * axiom of induction*.

- If
*K*is a set such that:- 0 is in
*K*, and - for every natural number
*n*,*n*being in*K*implies that*S*(*n*) is in*K*,

*K*contains every natural number. - 0 is in

The induction axiom is sometimes stated in the following form:

- If
*φ*is a unarypredicate such that:*φ*(0) is true, and- for every natural number
*n*,*φ*(*n*) being true implies that*φ*(*S*(*n*)) is true,

*φ*(*n*) is true for every natural number*n*.

In Peano's original formulation, the induction axiom is a