I don't really don't like the wording of the author - "infinity + 1" - because it isn't. The omegas are used to describe the ordering of the set and not, in any way, the "magnitude" of the infinity.

Well, I guess the catch phrase "infinity is a number - an infinite number of numbers" is just too good of a "deep wisdom" that people repeat it regardless of how much sense it makes.

Anyway, omega + 2 tells us, to put it simply, that we have a countably infinite subset, followed by a finite subset of two elements. In the same way, omega + omega would mean one countably infinite subset followed by another.

Note that this has no bearing whatsoever in the magnitude of the set. No matter how many omegas you add, you can still project the result onto the "basic" set of natural numbers. The labels don't matter, you aren't limited by the author's example of splitting even/odd. You can try it with sets {x, xx, xxx, ...} and {y, yy, yyy, ...} and they still project onto the naturals just fine.