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Topic: This Week's Finds in Mathematical Physics (Week 236)
Replies: 29   Last Post: Aug 24, 2006 9:00 AM

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john baez

Posts: 460
Registered: 12/6/04
Re: This Week's Finds in Mathematical Physics (Week 236)
Posted: Aug 2, 2006 11:21 PM
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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 http://www.cs.swan.ac.uk/~csetzer/index.html

Anton Setzer, Ordinal systems, in Sets and Proofs, Cambridge U. Press,
Cambridge, 2011, pp. 301-331. 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), 199-260. Also available at
http://www.cis.upenn.edu/~jean/gallier-old-pubs.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), 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,
http://homepages.inf.ed.ac.uk/v1phanc1/ordinal-notations.html

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




Date Subject Author
7/26/06
Read This Week's Finds in Mathematical Physics (Week 236)
baez@math.removethis.ucr.andthis.edu
7/27/06
Read Re: This Week's Finds in Mathematical Physics (Week 236)
david.corfield@tuebingen.mpg.de
7/29/06
Read Re: This Week's Finds in Mathematical Physics (Week 236)
baez@math.removethis.ucr.andthis.edu
7/30/06
Read Re: This Week's Finds in Mathematical Physics (Week 236)
david.corfield@tuebingen.mpg.de
8/1/06
Read Re: This Week's Finds in Mathematical Physics (Week 236)
john baez
7/28/06
Read Re: This Week's Finds in Mathematical Physics (Week 236)
Jim Heckman
7/29/06
Read Re: This Week's Finds in Mathematical Physics (Week 236)
baez@math.removethis.ucr.andthis.edu
7/31/06
Read Re: This Week's Finds in Mathematical Physics (Week 236)
Jim Heckman
8/1/06
Read Re: This Week's Finds in Mathematical Physics (Week 236)
Tom Leinster
8/2/06
Read Re: This Week's Finds in Mathematical Physics (Week 236)
john baez
7/28/06
Read Re: This Week's Finds in Mathematical Physics (Week 236)
tchow@lsa.umich.edu
7/29/06
Read Re: This Week's Finds in Mathematical Physics (Week 236)
baez@math.removethis.ucr.andthis.edu
7/30/06
Read Re: This Week's Finds in Mathematical Physics (Week 236)
tchow@lsa.umich.edu
8/2/06
Read Re: This Week's Finds in Mathematical Physics (Week 236)
john baez
8/7/06
Read Re: This Week's Finds in Mathematical Physics (Week 236)
Keith Ramsay
8/8/06
Read Re: This Week's Finds in Mathematical Physics (Week 236)
john baez
8/8/06
Read Re: This Week's Finds in Mathematical Physics (Week 236)
tchow@lsa.umich.edu
8/24/06
Read Re: This Week's Finds in Mathematical Physics (Week 236)
Nik Weaver
7/28/06
Read Re: Order-preserving embeddings of ordinals in the real numbers
David Madore
7/28/06
Read Re: Order-preserving embeddings of ordinals in the real numbers
G. A. Edgar
7/29/06
Read Re: Order-preserving embeddings of ordinals in the real numbers
baez@math.removethis.ucr.andthis.edu
7/30/06
Read graphical representation of epsilon_0 (was: Re: Order-preserving embeddings of ordinals in the real numbers)
David Madore
8/1/06
Read Pictures of infinity
baez@math.removethis.ucr.andthis.edu
8/2/06
Read Re: Pictures of infinity
tony@mantis.co.uk
8/2/06
Read Re: Pictures of infinity
Alec Edgington
8/3/06
Read Re: Pictures of infinity
Dave L. Renfro
7/28/06
Read Re: Order-preserving embeddings of ordinals in the real numbers
Ian A. Mason
7/28/06
Read Re: This Week's Finds in Mathematical Physics (Week 236)
Kevin Buzzard
7/29/06
Read Re: This Week's Finds in Mathematical Physics (Week 236)
baez@math.removethis.ucr.andthis.edu

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