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Applications of PCF theory

Published online by Cambridge University Press:  12 March 2014

Saharon Shelah
Saharon Shelah*
Affiliation:
Institute of Mathematics, The Hebrew University, Jerusalem, Israel Rutgers University, Department of Mathematics, New Brunswick, NJ, USA, E-mail:shelah@math.huji.ac.il
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Abstract

We deal with several pcf problems: we characterize another version of exponentiation: maximal number of k-branches in a tree with λ nodes, deal with existence of independent sets in stable theories, possible cardinalities of ultraproducts and the depth of ultraproducts of Boolean Algebras. Also we give cardinal invariants for each λ with a pcf restriction and investigate further TD{f). The sections can be read independently, although there are some minor dependencies.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 65 , Issue 4 , December 2000 , pp. 1624 - 1674
Copyright
Copyright © Association for Symbolic Logic 2000

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References

REFERENCES

[Bay]Bays, Timothy James, Multi-cardinal phenomena in stable theories, Ph.D. thesis, UCLA. 1994.Google Scholar
[CK]Chano, Chen C. and Keisler, Jerome H., Model theory, Studies in Logic and the Foundations of Mathematics, vol. 73, North Holland Publishing Co., Amsterdam, 1973.Google Scholar
[CuSh:541]Cummings, James and Shelah, Saharon, Cardinal invariants above the continuum, Annals of Pure and Applied Logic, vol. 75 (1995), pp. 251–2681.CrossRefGoogle Scholar
[For]Foreman, Matthew, Transfer properties, Ph.D. thesis, University of California, Berkeley, 1981.Google Scholar
[Kn]Kanamori, Akihiro, Weakly normal filters and irregular ultra-filters, Transactions of the American Mathematical Society, vol. 220 (1976), pp. 393–396.CrossRefGoogle Scholar
[LMSh: 198]Levinski, Jean Pierre, Magidor, Menachem, and Shelah, Saharon, Chang's conjecture for ℵω. Israel Journal of Mathematics, vol. 69 (1990), pp. 161–172.CrossRefGoogle Scholar
[MgSh:433]Magidor, Menachem and Shelah, Saharon, Length of Boolean algebras and ultraproducts, Mathematica Japonica, vol. 48 (1998), pp. 301–307.Google Scholar
[M]Monk, Donald, Cardinal functions of Boolean algebras, Progress in Mathematics, vol. 142, Birkhauser Verlag, Basel–Boston–Berlin, 1996.CrossRefGoogle Scholar
[RoSh:534]Roslanowski, Andrzej and Shelah, Saharon, Cardinal invariants of ultrapoducts of Boolean algebras, Fundamenta Mathematicae, vol. 155 (1998), pp. 101–151.CrossRefGoogle Scholar
[Sh:a]Shelah, Saharon, Classification theory and the number of nonisomorphic models, Studies in Logic and the Foundations of Mathematics, vol. 92, North-Holland Publishing Co., Amsterdam-New York, 1978.Google Scholar
[Sh:71]Shelah, Saharon, A note on cardinal exponentiation, this Journal, vol. 45 (1980), pp. 56–66.Google Scholar
[Sh: 111]Shelah, Saharon, On power of singular cardinals, Notre Dame Journal of Formal Logic, vol. 27 (1986), pp. 263–299.CrossRefGoogle Scholar
[Sh:c]Shelah, Saharon, Classification theory and the number of nonisomorphic models, Studies in Logic and the Foundations of Mathematics, vol. 92, North-Holland Publishing Co., Amsterdam, 1990.Google Scholar
[Sh:351]Shelah, Saharon, Reflecting stationary sets and successors of singular cardinals, Archive for Mathematical Logic, vol. 31 (1991), pp. 25–53.CrossRefGoogle Scholar
[Sh:355]Shelah, Saharon, ℵω+1 has a Jonsson algebra, Cardinal arithmetic (Gabbai, Dov M., Macintyre, Angus, and Scott, Dana, editors), Oxford Logic Guides, vol. 29, Oxford University Press, 1994.CrossRefGoogle Scholar
[Sh:g]Shelah, Saharon, Cardinal arithmetic, Oxford Logic Guides, vol. 29, Oxford University Press, 1994, on corrections and updates see http://front.math.ucdavis.edu/math.LO/9906022.CrossRefGoogle Scholar
[Sh:430]Shelah, Saharon, Further cardinal arithmetic, Israel Journal of Mathematics, vol. 95 (1996), pp. 61–114.CrossRefGoogle Scholar
[Sh:506]Shelah, Saharon, The pcf-theorem revisited, The mathematics of Paul Erdős, II (Graham, I. and Nešetřil, J., editors), Algorithms and Combinatorics, vol. 14, Springer, 1997, pp. 420–459.Google Scholar
[Sh:497]Shelah, Saharon, Set theory without choice: not everything on cofinality is possible, Archive for Mathematical Logic, vol. 36 (1997), pp. 81–125, A special volume dedicated to Professor Azriel Levy.CrossRefGoogle Scholar
[Sh:620]Shelah, Saharon, Special subsets ofcf(Μ)Μ, Boolean algebras and Maharam measure algebras, Topology and its Applications, vol. 99 (1999), pp. 135–235, General Topology and its Applications—Proceedings of 8th Prague Topological Symposium on general topology and its relations to modern analysis and algebra, part II 1996.CrossRefGoogle Scholar
[Sh:513]Shelah, Saharon, PCF and infinite free subsets, Archive for Mathematical Logic, (accepted).Google Scholar