transfinite induction

Suppose $\mathrm{\Phi }(\alpha )$ is a property defined for every ordinal $\alpha$, the principle of transfinite induction states that in the case where for every $\alpha$, if the fact that $\mathrm{\Phi }(\beta )$ is true for every implies that $\mathrm{\Phi }(\alpha )$ is true, then $\mathrm{\Phi }(\alpha )$ is true for every ordinal $\alpha$. Formally :

The principle of transfinite induction is very similar to the principle of finite induction, except that it is stated in terms of the whole class of the ordinals.

Title	transfinite induction
Canonical name	TransfiniteInduction
Date of creation	2013-03-22 12:29:03
Last modified on	2013-03-22 12:29:03
Owner	jihemme (316)
Last modified by	jihemme (316)
Numerical id	10
Author	jihemme (316)
Entry type	Theorem
Classification	msc 03B10
Synonym	principle of transfinite induction
Related topic	PrincipleOfFiniteInduction
Related topic	Induction
Related topic	TransfiniteRecursion