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**Yasen** My foremost problem is on which ordinal should I do induction?

If I do induction on $\displaystyle \alpha $, 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 $\displaystyle 0+ \beta < 0 + \gamma $, but the definition of addition with 0 is $\displaystyle \alpha + 0 = \alpha $ . 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 $\displaystyle \beta $, then sooner or later I would get $\displaystyle \beta > \gamma $, and then what?

Could you at least give me this hint?