Natural Numbers:Postulates

Peano's Axioms for the Natural Numbers

1. 1 is a natural number.
2. For every natural number $n$, the successor to $n$, ($n+1$), is also a natural number.
3. 1 is not a successor to any natural number.
4. If two numbers $n_{1}$ and $n_{2}$ have the same successor, then $n_{1}=n_{2}$.
5. If a set $S$ contains 1, and also contains the successor of every element $n$ in $S$, then every natural number is in $S$.

These axioms are used to build the set of natural numbers $\mathbb {N} =\{1,2,3,...,n,n+1,...\}$.