Date: Aug 2, 2006 11:21 PM
Author: john baez
Subject: Re: This Week's Finds in Mathematical Physics (Week 236)

Someday I'd like to go on and describe systems of notation for 
ordinals above the Feferman-Schuette ordinal. But I might not
get around to it, so here's some stuff for anyone interested -
including possibly my future self.

This paper introduced the "Schuette Klammersymbole", which
generalize the Veblen hierarchy:

27) Kurt Schuette, Kennzeichnung von Orgnungszahlen durch rekursiv
erklaerte Funktionen, Math. Ann 127 (1954), 15-32.

These papers discuss a general concept of "ordinal notation system",
which includes the Schuette Klammersymbole and also something
called the "n-ary Veblen hierarchy":

28) Anton Setzer, An introduction to well-ordering proofs in Martin-
Loef's type theory, in Twenty-Five Years of Constructive Type Theory,
eds. G. Sambin and J. Smith, Clarendon Press, Oxford, 1998, pp. 245-263.
Also available at

Anton Setzer, Ordinal systems, in Sets and Proofs, Cambridge U. Press,
Cambridge, 2011, pp. 301-331. Also available at

This paper has a nice expository section on generalizations of the
Veblen hierarchy:

29) Jean H. Gallier, What's so special about Kruskal's theorem and
the ordinal Gamma_0? A survey of some results in proof theory,
sec. 7: A glimpse at Veblen hierarchies, Ann. Pure Appl. Logic 53
(1991), 199-260. Also available at

This paper is very useful, since it compares different notations:

30) Larry W. Miller, Normal functions and constructive ordinal notations,
J. Symb. Log. 41 (1976), 439-459.

You can get it through JSTOR if you have access to that.

This webpage gives a nice definition of "ordinal notation system"
as a coalgebra of a certain functor - nice if you understand categories,
that is:

31) Peter Hancock, Ordinal notation systems,

Finally, the Wikipedia article on "large countable ordinals" has
some references to books which are, alas, out of print.