Patterns of structural reflection in the large-cardinal hierarchy
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- by Joan Bagaria and Philipp Lücke;
- Trans. Amer. Math. Soc. 377 (2024), 3397-3447
- DOI: https://doi.org/10.1090/tran/9120
- Published electronically: March 12, 2024
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Abstract:
We unveil new patterns of Structural Reflection in the large-cardinal hierarchy below the first measurable cardinal. Namely, we give two different characterizations of strongly unfoldable and subtle cardinals in terms of a weak form of the principle of Structural Reflection, and also in terms of weak product structural reflection. Our analysis prompts the introduction of the new notion of $C^{(n)}$-strongly unfoldable cardinal for every natural number $n$, and we show that these cardinals form a natural hierarchy between strong unfoldable and subtle cardinals analogous to the known hierarchies of $C^{(n)}$-extendible and $\Sigma _n$-strong cardinals. These results show that the relatively low region of the large-cardinal hierarchy comprised between the first strongly unfoldable and the first subtle cardinals is completely analogous to the much higher region between the first strong and the first Woodin cardinals, and also to the much further upper region of the hierarchy ranging between the first supercompact and the first Vopěnka cardinals.References
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Bibliographic Information
- Joan Bagaria
- Affiliation: ICREA (Institució Catalana de Recerca i Estudis Avançats), Spain; and Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Gran Via de les Corts Catalanes, 585, 08007 Barcelona, Catalonia, Spain
- MR Author ID: 340166
- ORCID: 0000-0002-4686-8222
- Email: joan.bagaria@icrea.cat
- Philipp Lücke
- Affiliation: Fachbereich Mathematik, Universität Hamburg, Bundesstraße 55, Hamburg 20146, Germany
- ORCID: 0000-0001-8746-5887
- Email: philipp.luecke@uni-hamburg.de
- Received by editor(s): October 13, 2022
- Received by editor(s) in revised form: November 6, 2023
- Published electronically: March 12, 2024
- Additional Notes: The first author was supported by the Generalitat de Catalunya (Catalan Government) under grant 2021 SGR 00348, and by the Spanish Government under grant MTM-PID2020-116773GBI00 and Europa Excelencia grant EUR2022-134032
The second author was supported by the Deutsche Forschungsgemeinschaft (Project number 522490605).
The research of both authors was supported by the Spanish Government under grant EUR2022-134032. Finally, this project had received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 842082 (Project SAIFIA: Strong Axioms of Infinity – Frameworks, Interactions and Applications). - © Copyright 2024 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 377 (2024), 3397-3447
- MSC (2020): Primary 03E55, 18A15, 03C55, 03E47
- DOI: https://doi.org/10.1090/tran/9120
- MathSciNet review: 4744784