I have a question here on ordinal arithmetic: Suppose ( not zero). What are ?

Am I right in assuming that the sum of two finite ordinals cannot be an infinite ordinal? If so I figure and can just be any finite ordinal. So and (by two rules in my notes).

Alpha and beta cannot be the other way round as right cancellation does not work.

So I think this proof is sound if the first statement is true. If anyone can comment on what I've done that'd be neat.