Return to Question

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, 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 $\epsilon_\beta$ 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 and call it something like $\phi_2(0)$.

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, Berkeley Logic Seminar, 3 October 2008.

After illustrating this notation for various ordinals up to $\epsilon_0$ and $\epsilon_1$, on page 13 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 $\epsilon_\beta$ 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 and call it something like $\phi_2(0)$.