Talk:Transfinite Induction

Generalization
This page only shows how to prove propositions for all ordinals. Generalizing to allow a lower bound is trivial and unimportant, but allowing an upper bound makes transfinite induction more general than finite induction. I believe the following statement should hold:

Let $l$ and $u$ be ordinals.

Suppose that


 * 1) $P(l)$
 * 2) For each ordinal $y$ such that $l \le y < u$,
 * if $P(x)$ for all ordinals $x$ such that $l \le x < y$ then $P(y)$.

Then $P(x)$ for each ordinal $x$ such that $l \le x < u$.

The case of finite induction corresponds to $u = \omega$. Dfeuer (talk) 16:40, 23 December 2012 (UTC)