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
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    $\begingroup$ In case the link to Jervell's notes goes dead, I've saved it at the Internet Archive: How to wellorder finite trees and get good ordinal notations $\endgroup$ Mar 11 at 14:17

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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    $\begingroup$ See also top p. 234 of An ordinal bound for the set of polynomial functions with exponentiation by Levitz (1978). FYI, I've known about this paper for about 25 years, it being among the several hundred "large countable ordinal papers" I've photocopied over the years (since the mid 1980s), and just now I came across your question/answer when looking for a link to something online (hopefully Wikipedia, but not found) that mentions the well-known "critical number" terminology specifically for $\epsilon$-numbers. I love your 3-part essay on large ordinals! $\endgroup$ Feb 28 at 20:49

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