Peano's postulates

The postulates isolated by Richard Dedekind (1831-1916) and formulated by the Italian mathematician G. Peano (1858-1932), that define the number series as the series of successors to the number zero. Informally they are: (i) zero is a number; (ii) zero is not the successor of any number; (iii) the successor of any number is a number; (iv) no two numbers have the same successor; and (v) if zero has a property, and if whenever a number has a property its successor has the property, then all numbers have the property. The fifth is the postulate justifying mathematical induction. It ensures that the series is closed, in the sense that nothing but zero and its successors can be numbers.

Any series satisfying such a set of axioms can be conceived as the sequence of natural numbers. Candidates from set theory include the Zermelo numbers, where the empty set is zero, and the successor of each number is its unit set, and the von Neumann numbers, where each number is the set of all smaller numbers.