Indecomposable Ordinals

Which ordinals are closed under addition? Which are closed under multiplication? Let us try to identify them exactly.

An interesting closure property

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?

Think about it…

Well, if an ordinal above ω is closed under addition, it would have to be bigger than ω + ω, and bigger than ω + ω + ω, and so forth. It would have to be bigger than ω · n for every finite number n. So it will be at least ω², the first compound limit ordinal.

Meanwhile, we can observe that ω² itself is closed under addition. Namely, if ordinals α, β are both below ω², then they are both below some ω · n for some large enough finite n, and consequently, we may bound the sum α + β as follows:

\( α + β\quad < \quad ω · n + ω · m\quad = \quadω · (n + m)\quad < \quad ω^2.\)

In other words, if α and β are smaller than ω², then α + β also is smaller than ω²—so the ordinal ω² is closed under addition. That is, ω² is additively indecomposable.

Additively Indecomposable Ordinals

In the general case, a nonzero ordinal λ is additively indecomposable if the sum of any two smaller ordinals remains smaller than λ. That is, α + β < λ whenever α, β < λ.

This is equivalent, it turns out, to λ being additively irreducible, meaning that it cannot be expressed as a sum λ = α + β of smaller ordinals α, β < λ. (The reader is asked to prove the equivalence in the questions for further thought.)

For example, we observed that the ordinal ω is additively indecomposable, since the sum of any two finite ordinals remains finite. But ω · 2 is not additively indecomposable, since ω + ω is not less than ω · 2. And we observed that ω² is the next additively indecomposable ordinal after ω.

Which ordinals exactly are additively indecomposable? Can we characterize them? And what about multiplicative indecomposability? Is that the same as multiplicative irreducibility? And what about exponential indecomposability?

Think about it...

Welcome to this essay on ordinal indecomposability, part of my essay series on the ordinals and ordinal arithmetic—you can find the other essays in the ordinal-arithmetic tag. After indecomposability, in the coming essays we shall get into the Cantor normal form and then the “natural” operations, from which the ordinals form a commutative semi-ring and generate what I call the natural ring of ordinals, sitting as a subring inside the surreal numbers and indeed inside the ominific integers. Thus, after building this foundation in the ordinals, we shall eventually return to my essay series on the surreal numbers, making use of our growing familiarity with the ordinals. You are welcome to join and follow along!

Let’s get into it…