Cardinal Arithmetic

Uri Abraham () and Menachem Magidor ()
Additional contact information
Uri Abraham: Ben Gurion University of the Negev, Department of Mathematics
Menachem Magidor: Hebrew University, Institute of Mathematics

Chapter 14 in Handbook of Set Theory, 2010, pp 1149-1227 from Springer

Abstract: 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.

Keywords: Regular Cardinal; Proper Ideal; Elementary Substructure; Singular Cardinal; Universal Sequence (search for similar items in EconPapers)
Date: 2010
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4020-5764-9_15

Ordering information: This item can be ordered from
http://www.springer.com/9781402057649

DOI: 10.1007/978-1-4020-5764-9_15

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().