If you add two finite numbers together, then the resulting sum, of course, remains finite. Thus, the finite numbers are closed under addition. We may view this phenomenon as expressing a closure property of the ordinal ω itself, the first infinite ordinal. Namely, ω is closed under addition—the sum of any two numbers smaller than ω, that is, the sum of any two finite numbers, has a result that is still below ω. This is what it means to say that the ordinal ω is additively indecomposable. Which other ordinals have this closure property? For example, what is the next additively indecomposable ordinal after ω?