Transfinite induction

If you are trying to prove that a property P holds for all ordinals then you can apply transfinite induction:

• Prove that P(0) holds true; and
• For any ordinal b, if P(a) is true for all ordinals a < b then P(b) is true as well.

The latter step is often broken down into two cases: the case for non-limit ordinals (ordinals which have an immediate predecessor), where the usual inductive approach can be applied (show that P(b) implies P(b+1)), and the case for limit ordinals[?], which have no predecessor, and thus cannot be handled by such an argument.

Typically, the case for limit ordinals is approached by noting that a limit ordinal b is (by definition) the union of all ordinals a < b and using this fact to prove P(b) assuming that P(a) holds true for all a < b.

The first step above is actually redundant. If P(b) follows from the truth of P(a) for all a < b, then it is simply a special case to say that P(0) is true, since it is vacuously true that P(b) holds for all b < 0.

An example here would be nice

[I agree that some examples should be added, but it's been a long time since I've looked at this material.]