Induction

Discipline: Philosophy

Also known as mathematical induction or finite induction, this affirms that to prove that a certain property P holds for all natural numbers, it suffices to show that P(l) is true and that P(k + 1) is true whenever P(k) is true.

Intuitively, one can think of this in terms of climbing a ladder, where proving P(l) corresponds to getting on the ladder, and proving P(k + 1) from P(k) corresponds to taking the (k + l)th step; once one knows how to move from one step of the ladder to the next, it is possible, in principle, to reach any step of the ladder.

There is an equivalent form of this principle, sometimes called complete induction, in which the inductive hypothesis above is replaced by the stronger assumption that P(j) is true for all integers j less than or equal to k; this latter form of induction is often simpler to apply in practice.

Also see: Goodman's paradox, inductivism, uniformity of nature, WELL-ORDERING PRINCIPLE

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