domino effect.

Mathematical induction can be informally illustrated by reference to the sequential effect of falling dominoes.

As an inference rule, mathematical induction can be justified as follows. Having proven the base case and the inductive step, then any value can be obtained by performing the inductive step repeatedly. It may be helpful to think of the domino effect. Consider a half line of dominoes each standing on end, and extending infinitely to the right. Suppose that:

1. The first domino falls right.
2. If a (fixed but arbitrary) domino falls right, then its next neighbor also falls right.

With these assumptions one can conclude (using mathematical induction) that all of the dominoes will fall right.

Mathematical induction, as formalized in the second-order axiom above, works because k is used to represent an arbitrary natural number. Then, using the inductive hypothesis, i.e. that P(k) is true, show P(k + 1) is also true. This allows us to "carry" the fact that P(0) is true to the fact that P(1) is also true, and carry P(1) to P(2), etc., thus proving P(n) holds for every natural number n.

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