
Re: This Week's Finds in Mathematical Physics (Week 236)
Posted:
Aug 2, 2006 11:21 PM


Someday I'd like to go on and describe systems of notation for ordinals above the FefermanSchuette 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), 1532.
These papers discuss a general concept of "ordinal notation system", which includes the Schuette Klammersymbole and also something called the "nary Veblen hierarchy": 28) Anton Setzer, An introduction to wellordering proofs in Martin Loef's type theory, in TwentyFive Years of Constructive Type Theory, eds. G. Sambin and J. Smith, Clarendon Press, Oxford, 1998, pp. 245263. Also available at http://www.cs.swan.ac.uk/~csetzer/index.html Anton Setzer, Ordinal systems, in Sets and Proofs, Cambridge U. Press, Cambridge, 2011, pp. 301331. Also available at http://www.cs.swan.ac.uk/~csetzer/index.html
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), 199260. Also available at http://www.cis.upenn.edu/~jean/gallieroldpubs.html
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), 439459.
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, http://homepages.inf.ed.ac.uk/v1phanc1/ordinalnotations.html
Finally, the Wikipedia article on "large countable ordinals" has some references to books which are, alas, out of print.

