 To illustrate the need for mathematical induction, assume that a property φ is true of the number zero and also that if true of a number then is true of its successor. Then, with our deductive systems, we can deduce that for any given number n, φ is true of it, φ(n). But we cannot deduce that φ is true of all numbers, ∀xφ(x); this inference step requires the rule of mathematical induction: