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A model for a very good scale and a bad scale

Published online by Cambridge University Press:  12 March 2014

Dima Sinapova
Dima Sinapova*
Affiliation:
Department of Mathematics, University of Californiaat Los Angeles, Los Angeles, CA 90095-1555, USA, E-mail: [email protected]
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Abstract

Given a supercompact cardinal κ and a regular cardinal λ < κ, we describe a type of forcing such that in the generic extension the cofinality of κ is λ, there is a very good scale at κ, a bad scale at κ, and SCH at κ fails. When creating our model we have great freedom in assigning the value of 2κ, and so we can make SCH hold or fail arbitrarily badly.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 73 , Issue 4 , December 2008 , pp. 1361 - 1372
Copyright
Copyright © Association for Symbolic Logic 2008

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References

REFERENCES

[1]Cummings, James and Foreman, Matthew, Marginalia to a theorem of Gitik and Sharon, talk.Google Scholar
[2]Cummings, James, Foreman, Matthew, and Magidor, Menachem, Squares, scales and stationary reflection, Journal of Mathematical Logic, vol. 1 (2001), pp. 35–98.CrossRefGoogle Scholar
[3]Gitik, Moti and Sharon, Assaf, On SCH and the approachahility property, Proceedings of the American Mathematical Society, vol. 136 (2008), pp. 311–320.CrossRefGoogle Scholar
[4]Jech, Thomas, Set theory, Springer Monographs in Mathematics, Springer-Verlag, 2003.Google Scholar
[5]Laver, Richard, Making the supercompactness of κ indestructible under κ-directed closed forcing, Israel Journal of Mathematics, vol. 29 (1978), no. 4, pp. 385–388.CrossRefGoogle Scholar
[6]Magidor, Menachem, Changing cofinality of cardinals, Fundamenta Mathematical vol. 99, no. 1, pp. 61–71.CrossRefGoogle Scholar
[7]Shelah, Saharon, Cardinal arithmetic, Oxford Logic Guides, vol. 29, Oxford University Press, 1994.CrossRefGoogle Scholar
[8]Solovay, Robert M., Reinhardt, William N., and Kanamori, Akihiro, Strong axioms of infinity and elementary embeddings, Annals of Mathematical Logic, vol. 13 (1978), no. 1, pp. 73–116.CrossRefGoogle Scholar