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THE UNIQUENESS OF ELEMENTARY EMBEDDINGS

Part of: Set theory

Published online by Cambridge University Press:  13 December 2024

GABRIEL GOLDBERG
GABRIEL GOLDBERG*
Affiliation:
UC BERKELEY, DEPARTMENT OF MATHEMATICS EVANS HALL, UNIVERSITY DRIVE BERKELEY, CA 94720 USA
*
E-mail: [email protected]
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Abstract

Much of the theory of large cardinals beyond a measurable cardinal concerns the structure of elementary embeddings of the universe of sets into inner models. This paper seeks to answer the question of whether the inner model uniquely determines the elementary embedding.

Keywords

large cardinalelementary embeddingstrongly compactUltrapower Axiom

MSC classification

Primary: 03E45: Inner models, including constructibility, ordinal definability, and core models 03E55: Large cardinals
Type
Article
Information
The Journal of Symbolic Logic , Volume 89 , Issue 4 , December 2024 , pp. 1430 - 1454
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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References

ReferenceS

Friedman, S.-D., Fuchino, S., and Sakai, H., On the set-generic multiverse , The Hyperuniverse Project and Maximality, Springer, Birkhäuser/Springer, Cham, 2018, pp. 109–124.CrossRefGoogle Scholar
Fuchs, G., Hamkins, J. D., and Reitz, J., Set-theoretic geology . Annals of Pure and Applied Logic, vol. 166 (2015), no. 4, pp. 464–501.CrossRefGoogle Scholar
Goldberg, G., The linearity of the Mitchell order . Journal of Mathematical Logic, vol. 18 (2018), no. 1, p. 1850005.CrossRefGoogle Scholar
Goldberg, G., The ultrapower axiom , Ph.D. thesis, Harvard University, 2019.CrossRefGoogle Scholar
Goldberg, G., The Ultrapower Axiom, ProQuest LLC, Ann Arbor, 2019.Google Scholar
Goldberg, G., The ultrapower axiom and the GCH . Journal of Mathematical Logic, vol. 21 (2020), p. 03.Google Scholar
Goldberg, G., Strongly compact cardinals and ordinal definability , Journal of Mathematical Logic, arXiv preprint, 2021, https://doi.org/10.48550/arXiv.2107.00513.CrossRefGoogle Scholar
Hamkins, J., Reitz, J., and Hugh Woodin, W., The ground axiom is consistent with  $V\ne HOD$ .Proceedings of the American Mathematical Society, vol. 136 (2008), no. 8, pp. 2943–2949.CrossRefGoogle Scholar
Kanamori, A., The Higher Infinite, 2nd ed., Springer Monographs in Mathematics, Springer, Berlin, 2009.Google Scholar
Ketonen, J., Strong compactness and other cardinal sins . Annals of Pure and Applied Logic, vol. 5 (1972/73), pp. 47–76.Google Scholar
Lévy, A. and Solovay, R. M., Measurable cardinals and the continuum hypothesis . Israel Journal of Mathematics, vol. 5 (1967), pp. 234–248.CrossRefGoogle Scholar
Reitz, J., The ground axiom . The Journal of Symbolic Logic, vol. 72 (2007), no. 4, pp. 1299–1317.CrossRefGoogle Scholar
Usuba, T., The downward directed grounds hypothesis and very large cardinals . Journal of Mathematical Logic, vol. 17 (2017), no. 2, Article no. 1750009, 24 pp.CrossRefGoogle Scholar
Usuba, T., Choiceless Löwenheim–Skolem property and uniform definability of grounds, Advances in Mathematical Logic, arXiv preprint, 2019, arXiv:1904.00895.Google Scholar