Peano's axioms for the Natural numbers

  1. 1 is a natural number.
  2. Every natural number nn has a natural number nn^\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 XX contains 1 and for every natural number nn also its successor nn^\prime, then the natural numbers form a subset of XX.
Peano later altered his axioms so that N\mathbb{N} included 0.