Abstract
Following the work of Gödel and Cohen we now know that it is impossible to determine the exact value of the continuum or of the power set of an arbitrary cardinal. Despite the limitations that the consistency methods impose, the quest for absolute answers as to the value of the power set (especially of singular cardinals) continues, and sometimes with surprising and unexpected directions. The pcf theory developed by Shelah and considerably extending previous work of Silver and Galvin and Hajnal and others is an important avenue in investigating set-theoretical and combinatorial questions about the power-set of singular cardinals. The aim of our chapter is to give a self-contained development of this theory and related results as well as some of its interesting applications. For example the celebrated theorem of Shelah that gives an absolute estimate on the number of countable subsets of ℵ ω (the first singular cardinal). The question whether this limitation can be improved is considered to be one of the main open problems in set theory.
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Bibliography
James E. Baumgartner and Karel L. Prikry. Singular cardinals and the Generalized Continuum Hypothesis. American Mathematical Monthly, 84:108–113, 1977.
Maxim R. Burke and Menachem Magidor. Shelah’s pcf theory and its applications. Annals of Pure and Applied Logic, 50(3):207–254, 1990.
Rysmard Engelking and Monica Karlowicz. Some theorems of set theory and their topological consequences. Fundamenta Mathematicae, 57:275–285, 1965.
John Gregory. Higher Souslin trees and the Generalized Continuum Hypothesis. The Journal of Symbolic Logic, 41(3):663–671, 1976.
András Hajnal, István Juhász, and Zoltán Szentmiklóssy. On CCC Boolean algebras and partial orders. Commentationes Mathematicae Universitatis Carolinae, 38(3):537–544, 1997.
Michael Holz, Karsten Steffens, and Edmund Weitz. Introduction to Cardinal Arithmetic. Birkhäuser, Boston, 1999.
Thomas J. Jech. Singular cardinals and the pcf theory. The Bulletin of Symbolic Logic, 1:408–424, 1995.
Thomas J. Jech. Set Theory. Springer, Berlin, 2002. Third millennium edition.
Akihiro Kanamori. The Higher Infinite. Springer Monographs in Mathematics Springer, Berlin, 2003. Corrected second edition.
Menachem Kojman. Exact upper bounds and their uses in set theory. Annals of Pure and Applied Logic, 92:267–282, 1998.
Kuneh Kunen. Set Theory, an Introduction to Independence Proofs. North-Holland, Amsterdam, 1988.
Azriel Levy. Basic Set Theory. Dover, New York, 2002. Revised edition.
S. Shelah. Diamonds. November 2007; Shelah’s list: 922.
S. Shelah. More on the revised GCH and middle diamond. Shelah’s list: 829.
Saharon Shelah. Cardinal Arithmetic. Oxford University Press, Oxford, 1994.
Saharon Shelah. Cellularity of free products of Boolean algebras (or topologies). Fundamenta Mathematicae, 166:153–208, 2000.
Saharon Shelah. The Generalized Continuum Hypothesis revisited. Israel Journal of Mathematics, 116:285–321, 2000.
Jack Silver. On the singular cardinals problem. In Proceedings of the International Congress of Mathematicians (Vancouver, BC, 1974), Vol. 1, pages 265–268. Canadian Mathematical Congress, Montreal, 1975.
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Abraham, U., Magidor, M. (2010). Cardinal Arithmetic. In: Foreman, M., Kanamori, A. (eds) Handbook of Set Theory. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-5764-9_15
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DOI: https://doi.org/10.1007/978-1-4020-5764-9_15
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