proof of principle of transfinite induction

To prove the transfinite inductionMathworldPlanetmath theoremMathworldPlanetmath, we note that the class of ordinalsMathworldPlanetmathPlanetmath is well-ordered by . So suppose for some Φ, there are ordinals α such that Φ(α) is not true. Suppose further that Φ satisfies the hypothesisMathworldPlanetmathPlanetmath, i.e. α(β<α(Φ(β))Φ(α)). We will reach a contradictionMathworldPlanetmathPlanetmath.

The class C={α:¬Φ(α)} is not empty. Note that it may be a proper classMathworldPlanetmath, but this is not important. Let γ=min(C) be the -minimal element of C. Then by assumptionPlanetmathPlanetmath, for every λ<γ, Φ(λ) is true. Thus, by hypothesis, Φ(γ) is true, contradiction.

Title proof of principle of transfinite induction
Canonical name ProofOfPrincipleOfTransfiniteInduction
Date of creation 2013-03-22 12:29:06
Last modified on 2013-03-22 12:29:06
Owner jihemme (316)
Last modified by jihemme (316)
Numerical id 11
Author jihemme (316)
Entry type Proof
Classification msc 03B10