Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

Is there any ordinal $\alpha$ such that $\omega ^ {\omega ^ \alpha} = \alpha$?

Could you please suggest me how to even try to solve this?

share|cite|improve this question
If $2^\alpha>\alpha$ (Cantor), then also $\omega^\alpha>\alpha$ and $\omega^{x}\geq x$ holds as well. – Peter Franek Jun 28 '14 at 7:43
@Peter: That's true for cardinal arithmetic; not for ordinal arithmetic. – Asaf Karagila Jun 28 '14 at 7:45
@AsafKaragila: I see; I don't understand it completely at this moment, so sorry for the confusion. – Peter Franek Jun 28 '14 at 7:48
@AsafKaragila: doesn't these identities hold for ordinal numbers? Can you have an $\alpha$ with $2^\alpha=\alpha$? – Peter Franek Jun 28 '14 at 7:53
@Peter: In ordinal arithmetic $2^\omega=\omega$. This is not the same exponentiation as cardinal exponentiation. See this post. – Asaf Karagila Jun 28 '14 at 7:56

Ordinals such that $\omega^{\alpha}=\alpha$ are called $\epsilon$-ordinals. The first such, $\epsilon$ zero is a tower of exponents, $$\epsilon_0=\omega^{\omega^{\omega^{\ddots}}}$$ (well I dont know how to make the diagonal dots go in the other direction)

It can be defined as follows $$\epsilon_0=\sup \beta_n$$

where $\beta_n$ is defined as

$$\beta_0=\omega \qquad \beta_{n+1}=\omega^{\beta_n}.$$

The epsilon ordinals $\epsilon_{\nu}$ form a closed unbounded set.

share|cite|improve this answer
Thanks for your answer! I've got the idea but I don't know how to prove that $sup \beta_n$ is an ordinal and that $\omega ^ \epsilon = \epsilon$ – Alex Jun 28 '14 at 12:50
Well supremum of a set of ordinals is again an ordinal, this is a standard result. for the equality use the fact that $\omega^{\alpha}$ is continous. – Rene Schipperus Jun 28 '14 at 12:55

HINT: What happens if $\alpha=\omega^\alpha$? Can we even have that?

share|cite|improve this answer

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.