Ordinal existence

Question

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

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

Answer 1

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

Answer 2

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.