# FOM: Shelah talk (cardinal powers) (fwd)

John Baldwin jbaldwin at math.uic.edu
Tue Aug 15 21:03:06 EDT 2000

Matt Frank and Joe Shipman asked about the state of cardinal arithmetic
in respect to Shelah's talk in LA.  I asked Andrzej Roslanowski
if he could give some references and they follow.

The second paper below explains the analogy between the late 19th
century replacement of polynomials by  ideals as a method for
solving problems of arithmetic and Shelah's idea of replacing
a set of possible cofinalities (pcf) for the exact cofinality as a
tool for studying cardinal arithmetic.

John Baldwin

Andrez's list of references follow.

>
> - the proof is fully presented in Shelah's book [Sh:g] (= Shelah, Saharon,
>   Cardinal Arithmetic, Oxford Logic Guides 29, General Editors: Dov
>   M. Gabbai, Angus Macintyre, Dana Scott, Oxford University Press 1994,
>   ISBN 0 19 853785 9). See Chapter IX there
>
> - Also a popular presentation of cardinal arithmetic was done in
> 	@article{Sh:400a,
> 	author = {Shelah, Saharon},
> 	journal = {{American Mathematical Society. Bulletin. New Series}},
> 	year = {1992},
> 	volume = {26},
> 	title = {{Cardinal arithmetic for skeptics}},
> 	pages = {197--210},
> (available from Rutgers, i.e., http://math.rutgers.edu/~shelah as well as
> from mathematics arXive, try e.g.
> http://front.math.ucdavis.edu/search/shelah+rn:400a)
>
> - current (more or less) updates on developments in the area of pcf theory,
> as well as some additional explanations and corrections/improvements are
> presented in
> 	@unpublished{Sh:E12,
> 	author = {Shelah, Saharon},
> 	title = {{Analytical Guide and Corrections to \cite{Sh:g}.}},},
> (again available from Rutgers; the version in mathematics arXive is not that
> much current, but I will update it soon I hope)
>
> - I do not remember what exactly was presented there, but one may look at
> the paper of Burke and Magidor:
> 	Burke, Maxim R.; Magidor, Menachem
>         Shelah's ${\rm pcf}$ theory and its applications.
>         Ann. Pure Appl. Logic 50 (1990), no. 3, 207--254.
> Also Menahem Kojman is writing an exposition of pcf (some parts are done, I
> think), but about that he should be contacted directly.
>
> - finally, a related "easy reading" could be
>         Holz, M.; Steffens, K.; Weitz, E.
>         Introduction to cardinal arithmetic.
>         Texts: Basel Textbooks] Birkhduser Verlag, Basel, 1999. viii+304pp.
>         ISBN: 3-7643-6124-7
> (it contains some elements of pcf)
> Yours, Andrzej
> --
> Andrzej Roslanowski
> Department of Mathematics        *  PHONE: +1-402-5543105
> University of Nebraska at Omaha  *  FAX:   +1-402-5542975
> Omaha, NE 68182-0243, USA        *  URL:   http://www.unomaha.edu/~aroslano
>