# Peano Axioms

**Peano Axioms** are axioms defining natural numbers set \(\mathbb N\) using set language. With \(+\) and \(\times\) defined by Peano Arithmetic, \((\mathbb N,+,0,\times,1)\) forms a commutative semiring.

#### Contents

## Axioms

The axioms are as follow. \(0\) is a symbol standing for a constant and \(S(\cdot)\) is a unary function named 'successor'.

- \(0 \in \mathbb N\)
- \(\forall x \in \mathbb N, S(x) \in \mathbb N\)
- \(x=y \Leftrightarrow S(x)=S(y)\)
- \(\not \exists x: S(x)=0\)
- \(\forall K \subseteq \mathbb N,(0 \in K) \wedge ((x \in K) \Rightarrow (S(x) \in K)) \Rightarrow K=\mathbb N\)

## Peano Arithmetic

Peano Arithmetic defines two binary functions in set \(\mathbb N\), namely **addition** \(+\) and **multiplication** \(\times\). They are defined recursively:

**Addition**

1. \(x+0=x\)

2. \(x+S(y)=S(x+y)\)

## Find \(S(S(0))+S(S(S(0)))\).

\(\;\;\;\;\color{red}{S(S(0))}+S(\color{red}{S(S(0))})\)

\(=S(\color{red}{S(S(0))}+S(\color{red}{S(0)}))\) (Apply (2))

\(=S(S(\color{red}{S(S(0))}+S(\color{red}0)))\) (Again)

\(=S(S(S(S(\color{red}{S(0)}+\color{red}0))))\) (And again)

\(=S(S(S(S(S(0)))))\) (Apply (1))And this is actually \(2+3=5\) in our normally used symbols.

Let \(S(\cdot)\) be the successor function in natural number set \(\mathbb N\). Compute \[ S(S(S(0)))+S(S(S(S(S(0))))) \; .\]

You may refer to wiki: Peano axioms.

From now on, for simplicity, we will use the normally used symbols to represent the natural numbers instead of a long series of \(S\).

**Multiplication**

1. \(x \times 0 = 0\)

2. \(x \times S(y) = x + (x \times y)\)

## Find \(2 \times 3\).

\(\;\;\;\;2 \times 3\)

\(=2+(2 \times 2)\)

\(=2+(2+(2 \times 1))\)

\(=2+(2+(2+(2 \times 0)))\)

\(=2+(2+(2+(0)))\)

\(=2+(2+(2))\)

\(=2+(4)\)

\(=6\)