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 $\phi_2(0)$.

  • 1
    $\begingroup$ 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 '16 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 '16 at 14:57
  • 1
    $\begingroup$ 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 '16 at 16:59
  • 1
    $\begingroup$ 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 '16 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 '16 at 5:52

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 $\epsilon_x = x$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.