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Veblen $\phi$ functions extend the $\xi \mapsto \phi(\xi) := \omega^\xi$ and the $\xi \mapsto \phi(1,\xi) := \varepsilon_\xi$ functions on the ordinals by repeatedly taking fixed points (I won't repeat the definitions, which are on Wikipedia). For a reason that isn't entirely clear to me, Veblen functions of just two variables seem to have gotten all the attention (to the point that the range $\phi(1,0,0)$ of the notation system they provide has a special name: the "Feferman-Schütte ordinal").

Essentially the only text I was able to find which discusses the Veblen functions of more than two variables is Veblen's original paper (published in 1908, "Continuous Increasing Functions of Finite and Transfinite Ordinals"), which is difficult to read: he starts his ordinals at $1$ and uses slightly different conventions on the $\phi$ functions than are now more common (e.g., on the order of the variables), and so on; more importantly, he does not discuss normal forms for ordinals produced by these functions.

So, is there a more modern account that I missed? Ideally one that would describe in some detail the system of ordinal notations that can be extracted from Veblen functions of $\omega$ variables (the "small" Veblen ordinal) or of the smallest fixed point of $\xi \mapsto \phi(1^{(\xi)},\ldots,0)$ (the "large" Veblen ordinal). But even a modern "translation" of Veblen's paper would be of some help.

Edit: There seems to be some kind of ordinal notation system related to Veblen functions of three variables (I think) in Ackermann's 1951 paper "Konstruktiver Aufbau eines Abschnitts der zweiten Cantorschen Zahlenklasse", though the notations are also very old (and the ordinals also start at $1$, apparently zero still hadn't been invented in 1951). I'll try to read it and see whether it's interesting.

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After some digging around, I found the following paper:

Kurt Schütte, “Kennzeichnung von Ordnungszahlen durch rekursiv erklärte Funktionen”, Math. Ann. 127 (1954), 15–32 (MR0060556)

The author describes Veblen functions (under a slightly different notation, which he calls “bracket symbol”) in a very clear way, explains how to use them to obtain a constructive system of ordinal notations (or a few closely related systems), and also explains the connection with Ackermann's system which is basically built from Veblen functions of three variables.

It's very well written and the proofs are given in full details. And despite the chronological proximity with Ackermann's paper, Schütte's has a much more “modern” feel to it (were it only for the fact that he starts his ordinals at $0$ and uses notations and terminology that seem more familiar to me). Maybe because the author was younger.

I also noticed Larry Miller's paper, “Normal Functions and Constructive Ordinal Notations”, J. Symbolic Logic 41 (1976), 439–459 (MR0409132), which is useful in that it connects different constructive ordinal notation systems, including Schütte's (hence, indirectly, Veblen functions).

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  • $\begingroup$ Hello! I'm trying to read it to establish a connection between Schutte's Klammersymbols and Veblen functions....but I don't know German. Do you by any chance have another reference for this? In particular, does the "Proof Theory" book by Schutte include what is discussed in the paper? Thanks! $\endgroup$
    – Guillermo Mosse
    Commented Mar 19, 2018 at 21:30
  • $\begingroup$ @SeñorBilly No, I'm pretty sure Schütte's Proof Theory has nothing about this. You could try Larry Miller's paper I mention at the end, but it has a bit too much in it so it's not very pleasant to read. I'm afraid I don't have any other suggestions (it was hard enough finding Schütte's paper already). $\endgroup$
    – Gro-Tsen
    Commented Mar 20, 2018 at 1:06
  • $\begingroup$ @GuillermoMosse In the paper given in Andreas Weiermann's answer, theorem 1.8 connects Klammersymbols to a "$\phi\langle\bullet\rangle$" function which is a coding of Bachmann's functions (1.9.(i) proves $\phi_0\begin{pmatrix}\xi_0&\ldots&\xi_n \\ \alpha_0&\ldots&\alpha_n\end{pmatrix}=\phi\langle\Omega^{\alpha_n}\xi_n+\ldots+\Omega^{\alpha_0}\xi_0\rangle$ for countable $\alpha_i,\xi_i$), then lemma 1.9 proves $\phi\langle\bullet\rangle$ satisfies the usual fixed point enumeration definition of the multi-variable Veblen function. This is only proven for Klammersymbols with countable entries. $\endgroup$
    – C7X
    Commented Jan 4 at 22:45
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I think the following paper by Buchholz should provide very useful information. http://www.mathematik.uni-muenchen.de/~buchholz/articles/jaegerfestschr_buchholz3.pdf

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