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# mathematical induction

Mathematical induction is a a specialized form of deductive reasoning used to prove a fact about all the elements in an infinite set by performing a finite number of steps.

In order for mathematical induction to work with an infinite set, that set must be denumerable, meaning that a one-to-one correspondence must exist between the elements of the set in question and the set of positive integers. In other words, it must be possible to express the set in the form an implied list of discrete elements such as {1, 2, 3, 4, ...}.

Consider a denumerably (also called countably) infinite set X with elements x1, x2, x3, x4, and so on. In order to prove a proposition about all the elements of X, we begin by proving that the proposition holds true for x1, the first element in the set X. Then we must prove that if the proposition holds true for some arbitrary element xn in X (where n is a positive integer), then the proposition also holds true for the next element xn+1 in set X. If we can do both of these two things successfully using deductive reasoning, we create an infinite chain of true statements by rigorous logical implication, proving the proposition true for all of the elements in X.

The first explicit formalization of the induction principle was composed by the French mathematician Blaise Pascal in 1665. Mathematical induction should not be confused with inductive reasoning. The former principle is mathematically rigorous (meaning that the conclusions are logically certain), but the latter methodology deals with probability and allows for some uncertainty.

This was last updated in June 2013
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