Peano's axioms for the Natural numbers

1. 1 is a natural number.
   
2. Every natural number $n$ has a natural number $n^\prime$ as a successor.
   
3. 1 is not the successor of a natural number.
   
4. Natural numbers with the same successor are the same.
   
5. If the set $X$ contains 1 and for every natural number $n$ also its successor $n^\prime$, then the natural numbers form a subset of $X$.

Peano later altered his axioms so that $\mathbb{N}$ included 0.