Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

As is well known, each natural number (except $0$) can be written uniquely as product of finitely many prime numbers (with $1$ being the empty product). My question is: Does some analogue theorem also hold for ordinal numbers?

share|cite|improve this question
Well, $\omega = 2 \times \omega$, so.... Also, unique prime factorization is up to order, and it's unclear what this means for ordinal multiplication, which is noncommutative. – Qiaochu Yuan Aug 24 '12 at 7:51
Good point. But what if we add the condition that the numbers must be multiplied in decreasing order? For natural numbers, this would not make a difference, and for infinite numbers, I believe it should resolve this specific problem. Or alternatively one could use the natural (Hessenberg) product. – celtschk Aug 24 '12 at 8:05
Well, $(\omega + 1) \times \omega = (\omega + 2) \times \omega$, so... – Qiaochu Yuan Aug 24 '12 at 8:17
Oh, right. So even with the ordering requirement, for the normal ordinal product, it cannot work. However, what about the natural (Hessenberg) product? – celtschk Aug 24 '12 at 9:03
This is a very interesting question! I believe the question you are asking is whether or not the ordinals form a UFD using natural sum and product. The ordinals below $\omega^\omega$ endowed with natural sum and product are isomorphic to polynomials with natural number coefficients, so they form a UFD. I believe this remains true for higher ordinals, but I don't have a proof. – Deedlit Sep 23 '12 at 10:18
up vote 1 down vote accepted

Theorem 3 of Shinpei Oka's On Telgárski's formula (online summary) offers an affirmative answer.

share|cite|improve this answer
Sierpinski's book Cardinal and Ordinal Numbers is pretty much the bible for this kind of stuff. I don't have my copy with me where I'm at right now to give any suggestions based on it, but since no one else has yet mentioned Sierpinski's book, I thought I'd mention it in case the OP doesn't know about it. – Dave L. Renfro Jun 20 '14 at 14:10

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.