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, we define that an ordinal λ is additively indecomposable if every finite sum of ordinals below λ remains below λ. The ordinal ω, for example, is additively indecomposable, since the sum of finitely many finite numbers is finite, and we just observed above that ω² is additively indecomposable. The number 1 also is additively indecomposable, since every finite sum of ordinals below 1 adds up to 0, which remains less than 1.

What about 0 itself? Is 0 additively indecomposable? One might be inclined to say that 0 is additively indecomposable in a vacuous manner, since there are no smaller ordinals and therefore no finite sums of smaller ordinals. But this is not actually quite right according to the letter of the definition in light of the empty sum, which after all is a finite sum of ordinals, vacuously all less than 0, but the empty sum has value 0, which is not less than 0. On this technicality, therefore, 0 does not officially count as additively indecomposable. But actually, this is the answer we shall ultimately want, for this outcome makes for a smoother theory overall. The situation is similar to the question in number theory of whether the number 1 counts as prime. Sure, the only factors are 1 and itself, and yet mathematicians have agreed that we should not count 1 as prime. The number 1 after all is the value of the empty product, which is vacuously a product of smaller numbers. So 1 can be factored as a product of smaller numbers, the empty product, just as 0 is the value of a finite sum of smaller numbers, the empty sum.

Meanwhile, this consideration about the empty sum can also simply be absorbed into the statement by defining equivalently that an ordinal λ is additively indecomposable if 0 < λ and α + β < λ whenever α, β < λ. The stipulation that 0 < λ handles the empty sum, and the other finite sums are generated from ordinals below λ by adding extra terms one at a time. And so one often sees this latter definition, avoiding any need to consider the empty sum.

The additive indecomposability of an ordinal λ turns out to be equivalent to λ being additively irreducible, meaning that it cannot be expressed as a finite sum of smaller ordinals. Since 0 is the empty sum, an ordinal λ is additively irreducible if and only if 0 < λ and it is impossible that λ = α + β for some α, β < λ. (The reader is asked to prove the equivalence of additive indecomposability and irreducibility in the questions for further thought.)

Characterizing additive indecomposability

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 omnific 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…