Jervell has a notation for countable ordinals up to the small Veblen ordinal using trees:

• Herman Ruge Jervell, [How to wellorder finite trees
and get good ordinal notations](http://www.mittag-leffler.se/sites/default/files/IML-0001-32.pdf), Berkeley Logic Seminar, 3 October 2008.

After illustrating this notation for various ordinals up to $\epsilon_0$ and $\epsilon_1$, he illustrates it for two ordinals that he calls $\kappa_1$ and $\kappa_\omega$.  He calls them 'critical $\epsilon$-numbers'.  **What are these ordinals?**

I'll make a wild guess: $\kappa_\alpha$ is the $\alpha$th solution of the equation

$$ \beta = \epsilon_\beta $$

where the [epsilon number](https://en.wikipedia.org/wiki/Epsilon_numbers_(mathematics)) $\epsilon_\beta$ is, in turn, the $\beta$th solution of the equation

$$ \gamma = \omega^\gamma.$$

Am I right?

Separately: **how commonly used is this notation $\kappa_\alpha$ for certain countable ordinals?**  I've never seen it anywhere else.  Usually when people hit the first solution of $ \beta = \epsilon_\beta $ they introduce the [Veblen hierarchy](https://en.wikipedia.org/wiki/Veblen_function#The_Veblen_hierarchy) and call it something like $\phi_2(0)$.