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Guessing models imply the singular cardinal hypothesis


Author: John Krueger
Journal: Proc. Amer. Math. Soc. 147 (2019), 5427-5434
MSC (2010): Primary 03E05; Secondary 03E40
DOI: https://doi.org/10.1090/proc/14739
Published electronically: August 7, 2019
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Abstract: In this article we prove three main theorems: (1) guessing models are internally unbounded, (2) for any regular cardinal $ \kappa \ge \omega _2$, $ \mathsf {ISP}(\kappa )$ implies that $ \mathsf {SCH}$ holds above $ \kappa $, and (3) forcing posets which have the $ \omega _1$-approximation property also have the countable covering property. These results solve open problems of Viale [Ann. Pure Appl. Logic 163 (2012), no. 11, 1660-1678] and Hachtman and Sinapova [J. Symb. Log. 84 (2019), no. 2, 713-725].


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Additional Information

John Krueger
Affiliation: Department of Mathematics, University of North Texas, 1155 Union Circle #311430, Denton, Texas 76203
Email: jkrueger@unt.edu

DOI: https://doi.org/10.1090/proc/14739
Keywords: ISP, SCH, guessing model, internally unbounded, approximation property, countable covering property.
Received by editor(s): March 25, 2019
Published electronically: August 7, 2019
Additional Notes: This material is based upon work supported by the National Science Foundation under Grant No. DMS-1464859
Communicated by: Heike Mildenberger
Article copyright: © Copyright 2019 American Mathematical Society