0 is a natural number.
if is a natural number, then (its increment) is also a natural number.
0 is not the successor of any natural number; i.e., we have for every natural number .
different natural numbers must have different successors: i.e., if are natural numbers and , then . equivalently, if then we must have .
let be any property pertaining to a natural number . suppose that is true, and suppose that whenever is true, is also true. then is true for every natural number .