Proof of Mathematical Induction
The principle of mathematical induction is usually stated as an axiom of the natural numbers; see Peano axioms. However, it can be proved in some logical systems. For instance, it can be proved if one assumes:
- The set of natural numbers is well-ordered.
- Every natural number is either zero, or n+1 for some natural number n.
- For any natural number n, n+1 is greater than n.
To derive simple induction from these axioms, we must show that if P(n) is some proposition predicated of n, and if:
- P(0) holds and
- whenever P(k) is true then P(k+1) is also true
then P(n) holds for all n.
Proof. Let S be the set of all natural numbers for which P(n) is false. Let us see what happens if we assert that S is nonempty. Well-ordering tells us that S has a least element, say t. Moreover, since P(0) is true, t is not 0. Since every natural number is either zero or some n+1, there is some natural number n such that n+1=t. Now n is less than t, and t is the least element of S. It follows that n is not in S, and so P(n) is true. This means that P(n+1) is true, and so P(t) is true. This is a contradiction, since t was in S. Therefore, S is empty.
It can also be proved that induction, given the other axioms, implies well-ordering.
Read more about this topic: Mathematical Induction
Famous quotes containing the words proof of, proof, mathematical and/or induction:
“There is no better proof of a man’s being truly good than his desiring to be constantly under the observation of good men.”
—François, Duc De La Rochefoucauld (1613–1680)
“There are some persons in this world, who, unable to give better proof of being wise, take a strange delight in showing what they think they have sagaciously read in mankind by uncharitable suspicions of them.”
—Herman Melville (1819–1891)
“It is by a mathematical point only that we are wise, as the sailor or the fugitive slave keeps the polestar in his eye; but that is sufficient guidance for all our life. We may not arrive at our port within a calculable period, but we would preserve the true course.”
—Henry David Thoreau (1817–1862)
“One might get the impression that I recommend a new methodology which replaces induction by counterinduction and uses a multiplicity of theories, metaphysical views, fairy tales, instead of the customary pair theory/observation. This impression would certainly be mistaken. My intention is not to replace one set of general rules by another such set: my intention is rather to convince the reader that all methodologies, even the most obvious ones, have their limits.”
—Paul Feyerabend (1924–1994)