could you help me with the following?
Prove by transfinite induction that , where are ordinals .
The way we defined transfinite induction is:
Suppose that is a statement and
3. If is a limit and , , then
Then for all ordinals ,
If I do induction on , which seems the most natural choice, I face the problem that ordinal addition is defined "on the right", and is moreover not commutative.
(for example, the base case would be that , but the definition of addition with 0 is . This could be circumvented, but I think it will prove more difficult to do so in the later cases).
On the other hand, if I do induction on , then sooner or later I would get , and then what?
Could you at least give me this hint?