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 $\phi_2(0)$.

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    $\begingroup$ There might be one or two tags that are also relevant, maybe ordinal-analysis or some other proof theory related tag. $\endgroup$
    – Asaf Karagila
    Jul 2, 2016 at 9:25
  • $\begingroup$ Thanks. I added ordinal analysis, mainly because experts in that may know the answer to this question. $\endgroup$
    – John Baez
    Jul 2, 2016 at 14:57
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    $\begingroup$ Are you sure you linked to the right paper? The PDF is not searchable so I might have missed it, but I couldn't find any occurrence of the word "critical" or of $\kappa_1$ or $\kappa_\omega$ in it. • Incidentally, if you're looking for good references on ordinal notations up to the small and large Veblen ordinals, I recommend Schütte's paper (see here). Beyond that, it gets harder. $\endgroup$
    – Gro-Tsen
    Jul 2, 2016 at 16:59
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    $\begingroup$ I also can't find it in the paper, but your guess is very plausible: I would call the critical $\varepsilon$-numbers the ordinals of the form $\varphi(2,\alpha)$. For your other question: I personally have never seen the notation $\kappa_\alpha$ for a countable ordinal that I recall (and I've looked at a fair bit of proof-theoretical literature). $\endgroup$ Jul 2, 2016 at 18:09
  • $\begingroup$ Aargh, sorry, I linked to the wrong PDF! I fixed the link; it's on page 13. I should also try to understand Jervell's tree notation well enough that I can simply figure out what he means by $\kappa_\alpha$. $\endgroup$
    – John Baez
    Jul 3, 2016 at 5:52

1 Answer 1


I think I've blundered into an answer to my own question. In this paper:

• Hilbert Levitz, Transfinite ordinals and their notations: for the uninitiated.

the author writes:

The first critical epsilon number is defined as follows. Arrange the solutions of $\omega^x = x$ in order and call them $\epsilon_0, \epsilon_1, \epsilon_2, \dots$ etc. Then the first critical epsilon number is the smallest member of the sequence equal to own subscript.

Since Jervell calls the numbers $\kappa_\alpha$ "critical $\epsilon$-numbers", I conclude that he's probably talking about the same concept: $\kappa_\alpha$ is the $\alpha$th solution of $\epsilon_x = x$.


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