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WEAK SQUARES AND VERY GOOD SCALES

Published online by Cambridge University Press:  30 January 2018

MAXWELL LEVINE
MAXWELL LEVINE*
Affiliation:
KURT GÖDEL RESEARCH CENTER UNIVERSITY OF VIENNA VIENNA, AUSTRIAE-mail:[email protected]
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Abstract

We assume the existence of a supercompact cardinal and produce a model with weak square but no very good scale at a particular cardinal. This follows work of Cummings, Foreman, and Magidor, but uses a different approach. We produce another model, starting from countably many supercompact cardinals, where □K,<K holds but □K, λ fails for λ < K.

Keywords

03E3503E55large cardinalsforcing
Type
Articles
Information
The Journal of Symbolic Logic , Volume 83 , Issue 1 , March 2018 , pp. 1 - 12
Copyright
Copyright © The Association for Symbolic Logic 2018 

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References

REFERENCES

Cummings, J., Iterated forcing and elementary embeddings, Handbook of Set Theory (Foreman, M. and Kanamori, A., editors), Springer, New York, 2010, pp. 775–883.Google Scholar
Cummings, J., Foreman, M., and Magidor, M., Squares, scales, and stationary reflection. Journal of Mathematical Logic, vol. 1 (2001), pp. 35–98.Google Scholar
Cummings, J. and Schimmerling, E., Indexed squares. Israel Journal of Mathematics, vol. 131 (2002), pp. 61–99.Google Scholar
Eisworth, T., Successors of singular cardinals, Handbook of Set Theory (Foreman, M. and Kanamori, A., editors), Springer, 2010, pp. 1229–1350.CrossRefGoogle Scholar
Gitik, M. and Sharon, A., On SCH and the approachability property. Proceedings of the American Mathematical Society, vol. 136 (2008), no. 1, pp. 311–320.Google Scholar
Jensen, R., The fine structure of the constructible hierarchy. Annals of Mathematical Logic, vol. 4 (1972), pp. 229–308.Google Scholar
Laver, R., Making the supercompactness of κ indestructible under κ-directed closed forcing. Israel Journal of Mathematics, vol. 29 (1978), pp. 385–388.CrossRefGoogle Scholar
Levine, M., Stationary sets added when forcing square, submitted, 2016.Google Scholar
Magidor, M., Reflecting stationary sets, this Journal, vol. 47 (1982), no. 4, pp. 775–771.Google Scholar
Schimmerling, E., Combinatorial principles in the core model for one Woodin cardinal. Annals of Pure and Applied Logic, vol. 74 (1995), pp. 153–201.Google Scholar
Sharon, A., Weak squares, scales, stationary reflection, and the failure of SCH. Ph.D. thesis, Tel Aviv University, 2005.Google Scholar
Shelah, S., Cardinal Arithmetic, Oxford Logic Guides, vol. 29, Oxford University Press, London and New York, 1994.Google Scholar
Sinapova, D. and Unger, S., Combinatorics at ${\aleph _\omega }$. Annals of Pure and Applied Logic, vol. 165 (2014), pp. 996–1007.CrossRefGoogle Scholar