principle of finite induction

The principle of finite induction, also known as mathematical induction, is commonly formulated in two ways. Both are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath. The first formulation is known as weak inductionMathworldPlanetmath. It asserts that if a statement P(n) holds for n=0 and if P(n)P(n+1), then P(n) holds for all natural numbersMathworldPlanetmath n. The case n=0 is called the base case or base step and the implicationMathworldPlanetmath P(n)P(n+1) is called the inductive step. In an inductive proof, one uses the term induction hypothesis or inductive hypothesis to refer back to the statement P(n) when one is trying to prove P(n+1) from it.

The second formulation is known as strong or completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath induction. It asserts that if the implication n((m<nP(m))P(n)) is true, then P(n) is true for all natural numbers n. (Here, the quantifiersMathworldPlanetmath range over all natural numbers.) As we have formulated it, strong induction does not require a separate base case. Note that the implication n((m<nP(m))P(n) already entails P(0) since the statement m<0P(m) holds vacuously (there are no natural numbers less that zero).

A moment’s thought will show that the first formulation (weak induction) is equivalent to the following:

Let S be a set natural numbers such that

  1. 1.

    0 belongs to S, and

  2. 2.

    if n belongs to S, so does n+1.

Then S is the set of all natural numbers.

Similarly, strong induction can be stated:

If S is a set of natural numbers such that n belongs to S whenever all numbers less than n belong to S, then S is the set of all natural numbers.

The principle of finite induction can be derived from the fact that every nonempty set of natural numbers has a smallest element. This fact is known as the well-ordering principle for natural numbers. (Note that this is not the same thing as the well-ordering principle, which is equivalent to the axiom of choiceMathworldPlanetmath and has nothing to do with induction.)

Title principle of finite induction
Canonical name PrincipleOfFiniteInduction
Date of creation 2013-03-22 11:46:41
Last modified on 2013-03-22 11:46:41
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 24
Author CWoo (3771)
Entry type TheoremMathworldPlanetmath
Classification msc 03E25
Classification msc 00-02
Related topic TransfiniteInduction
Related topic AnExampleOfMathematicalInduction
Related topic Induction
Related topic WellFoundedInduction
Defines induction hypothesis
Defines inductive hypothesis
Defines base case
Defines base step