- There is a natural number 0.
- Every natural number a has a successor, denoted by a + 1.
- There is no natural number whose successor is 0.
- Distinct natural numbers have distinct successors: if a <> b, then a + 1 <> b + 1.
- If a property is possessed by 0 and and also by the successor of every natural number it is possessed by, then it is possessed by all natural numbers.
These axioms are sometimes paraphrased differently, starting at 1 instead of 0.