However, I've always been fascinated with the big topics of logic and set theory, and so I'm now working my way through Kunen's book, including the exercises.

And I got a little hang-up there now, in the first chapter exercise 5 (yes, I'm rusty).

Here's the objective: Let a be a limit ordinal. Show that the following are equivalent:

(A) F.a. b, c < a: b+c<a

(B) F.a. b<a: b+a=a

(C) F.a. X subset of a: type(X) = a or type(a\X) = a

(D) a = omega

^{delta}for some ordinal delta

The bit I'm having issues with here is (C) - I've gotten the equivalence for the (A), (B) and (D). And while I'm certain it's really just a stupid little thing I'm overlooking, I just don't see it. My initial thought was to work via type(X) + type(a\X), but that sum can (err, note the current forum madness, that's a "c a n") be < a even for limit ordinals (consider X the final omega segment of omega

^{2}+ omega) - I believe "=" only happens for indecomposable ordinals - i.e. the ones of this type.

Note that Cantor normal form is the next exercise (to which this one is meant to lead), so that can't be used. Any pointers?