Peano & Arithmetic Axioms

Peano & Arithmetic Axioms

This is a formal set of axioms defining the natural numbers and their arithmetic operations. 1. Zero is a natural number
$0\in \mathbb{N}$ 2. Every natural number has a unique successor
$\forall n\in \mathbb{N},S(n)\in \mathbb{N}$ 3. Zero is not the successor of any number
$\forall n\in \mathbb{N},S(n)\ne 0$ 4. Distinct numbers have distinct successors
$\forall m,n\in \mathbb{N},S(m)=S(n)⟹m=n$ 5. Principle of mathematical induction
If a set contains 0 and is closed under the successor function, then it contains all natural numbers. 6. Addition is associative
$\forall a,b,c\in \mathbb{N},(a+b)+c=a+(b+c)$ 7. Addition is commutative
$\forall a,b\in \mathbb{N},a+b=b+a$ 8. Multiplication is associative
$\forall a,b,c\in \mathbb{N},(a\cdot b)\cdot c=a\cdot (b\cdot c)$ 9. Multiplication distributes over addition
$\forall a,b,c\in \mathbb{N},a\cdot (b+c)=a\cdot b+a\cdot c$