Peano Axioms

Definition

Axioms for the Natural Number

Axioms

$\exists 1\in \mathbb{N}$

1 as a natural number

$\forall n\in \mathbb{N},\exists n^{\prime }\in \mathbb{N}$

For every natural number $n$, the successor $n^{\prime }$ is a natural number

$\nexists n\in \mathbb{N},n^{\prime }=1$

For every natural number $n$, $n^{\prime }=1$ is false. There is no natural number whose successor is 1

$\forall m,n\in \mathbb{N},m^{\prime }=n^{\prime }\Rightarrow m=n$

For all natural numbers $m,n$, if $m^{\prime }=n^{\prime }$, then $m=n$

$\forall S\subset \mathbb{N},(1\in S)\wedge (\forall n,(n\in S)\Rightarrow (n^{\prime }\in S))\Rightarrow S=\mathbb{N}$

If $S$ is a set such that: $0\in S$ and for every natural number $n$, $n$ being in $S$ implies that $n^{\prime }$ is in $S$, then $S$ contains every natural number.