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.

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.