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:
“A short letter to a distant friend is, in my opinion, an insult like that of a slight bow or cursory salutation—a proof of unwillingness to do much, even where there is a necessity of doing something.”
—Samuel Johnson (1709–1784)
“He who has never failed somewhere, that man can not be great. Failure is the true test of greatness. And if it be said, that continual success is a proof that a man wisely knows his powers,—it is only to be added, that, in that case, he knows them to be small.”
—Herman Melville (1819–1891)
“The most distinct and beautiful statement of any truth must take at last the mathematical form.”
—Henry David Thoreau (1817–1862)
“They relieve and recommend each other, and the sanity of society is a balance of a thousand insanities. She punishes abstractionists, and will only forgive an induction which is rare and casual.”
—Ralph Waldo Emerson (1803–1882)