Talk:Transfinite Induction/Principle 1

The validity problem

To ward off a revert, the problem with what was here before is that it did not properly justify the existence of a minimal element of $\operatorname{On} \setminus A$.

The fact it needs for this, specifically, is that the epsilon relation is strongly well-founded on $\operatorname{On}$. I imagine there are any number of ways to prove this (the most trivial being to accept AoF, in which case Axiom of Foundation (Strong Form) does the trick), but it is not inherently trivial enough to handwave. --Dfeuer (talk) 20:34, 24 April 2013 (UTC)

Statement

I think it is valuable to add the statement about all smaller ordinals—that is precisely what makes this induction. I had to stare at the statement several times before I realized that's what it was getting at, and that shouldn't be necessary. --Dfeuer (talk) 18:41, 25 April 2013 (UTC)