proof of the well-founded induction principle

This proof is very similar to the proof of the transfinite inductionMathworldPlanetmath theorem. Suppose Φ is defined for a well-founded set (S,R), and suppose Φ is not true for every aS. Assume further that Φ satisfies requirements 1 and 2 of the statement. Since R is a well-founded relation, the set {aS:¬Φ(a)} has an R minimal element r. This element is either an R minimal element of S itself, in which case condition 1 is violated, or it has R predessors. In this case, we have by minimality Φ(s) for every s such that sRr, and by condition 2, Φ(r) is true, contradictionMathworldPlanetmathPlanetmath.

Title proof of the well-founded induction principle
Canonical name ProofOfTheWellfoundedInductionPrinciple
Date of creation 2013-03-22 12:42:20
Last modified on 2013-03-22 12:42:20
Owner jihemme (316)
Last modified by jihemme (316)
Numerical id 7
Author jihemme (316)
Entry type Proof
Classification msc 03B10