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# Peano arithmetic

*Peano’s axioms* are a definition of the set of natural numbers, denoted $\mathbb{N}$. From these axioms *Peano arithmetic* on natural numbers can be derived.

1. $0\in\mathbb{N}$ (0 is a natural number)

2. For each $x\in\mathbb{N}$, there exists exactly one $x^{{\prime}}\in\mathbb{N}$, called the

*successor*of $x$3. $x^{{\prime}}\neq 0$ (0 is not the successor of any natural number)

4. $x=y$ if and only if $x^{{\prime}}=y^{{\prime}}$.

5. (

*axiom of induction*) If $M\subseteq\mathbb{N}$ and $0\in M$ and $x\in M$ implies $x^{{\prime}}\in M$, then $M=\mathbb{N}$.

The *successor* of $x$ is sometimes denoted $Sx$ instead of $x^{{\prime}}$. We then have $1=S0$, $2=S1=SS0$, and so on.

Peano arithmetic consists of statements derived via these axioms. For instance, from these axioms we can define addition and multiplication on natural numbers. Addition is defined as

$\displaystyle x+1$ | $\displaystyle=$ | $\displaystyle x^{{\prime}}\quad\text{for all }x\in\mathbb{N}$ | ||

$\displaystyle x+y^{{\prime}}$ | $\displaystyle=$ | $\displaystyle(x+y)^{{\prime}}\quad\text{for all }x,y\in\mathbb{N}$ |

Addition defined in this manner can then be proven to be both associative and commutative.

Multiplication is

$\displaystyle x\cdot 1$ | $\displaystyle=$ | $\displaystyle x\quad\text{for all }x\in\mathbb{N}$ | ||

$\displaystyle x\cdot y^{{\prime}}$ | $\displaystyle=$ | $\displaystyle x\cdot y+x\quad\text{for all }x,y\in\mathbb{N}$ |

This definition of multiplication can also be proven to be both associative and commutative, and it can also be shown to be distributive over addition.

## Mathematics Subject Classification

03F30*no label found*

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## Comments

## use of 0 as the start of the natural numbers.

Peano originally didn't use 0 as the start of the Natural numbers. He used 1 in axioms 1 and 3. The use of 0 as the beginning of the "natural" numbers is a set theoretic concept, and as such is not "natural" in the least. The idea of zero as a number took ages to develop, and is anything but natural.

Also, in Peano Arithmetic, you have to define 1 as the successor of 0 before you can use it and there is no justification from the axioms for using it in the definitions of addition and multiplication rather than something else. But by starting with 1 as the first natural number, the definitions follows from the axioms as 1 generates all the natural numbers by successive addition, which is something 0 doesn't do.

I would prefer that you call the Axioms in this section The Extended Paeno Axioms, since these are not what he wrote in his papers.